Schrödinger equation
In physics, especially quantum mechanics, the Schrödinger equation is an equation that describes how the quantum state of a physical system changes in time. It is as central to quantum mechanics as Newton's laws are to classical mechanics.
In the standard interpretation of quantum mechanics, the quantum state, also called a wavefunction or state vector, is the most complete description that can be given to a physical system. Solutions to Schrödinger's equation describe atomic and subatomic systems, electrons and atoms, but also macroscopic systems, possibly even the whole universe. The equation is named after Erwin Schrödinger who discoved it in 1926.[1]
Schrodinger's equation can be mathematically transformed into the Heisenberg formalism, and into the Feynman path integral. The Schrödinger equation describes time in a way that is inconvenient for relativistic theories, a problem which is less severe in Heisenberg's formulation and completely absent in the path integral.
[edit] Historical background and development
-
Einstein interpreted Planck's quanta as photons, particles of light, and proposed that the energy of a photon is proportional to its frequency, a mysterious wave-particle duality. Since energy and momentum are related in the same way as frequency and wavenumber in relativity, it followed that the momentum of a photon is proportional to its wavenumber.
DeBroglie hypothesized that this is true for all particles, for electrons as well as photons, that the energy and momentum of an electron are the frequency and wavenumber of a wave. Assuming that the waves travel roughly along classical paths, he showed that they form standing waves only for certain discrete frequencies, discrete energy levels which reproduced the old quantum condition.
Following up on these ideas, Schrödinger decided to find a proper wave equation for the electron. He was guided by Hamilton's analogy between mechanics and optics, encoded in the observation that the zero-wavlength limit of optics resembles a mechanical system--- the trajectories of light rays become sharp tracks which obey a principle of least action. Hamilton believed that mechanics was the zero-wavelength limit of wave propagation, but did not formulate an equation for those waves. This is what Schrödinger did, and a modern version of his reasoning is contained in the box below.
Long derivation (advanced)
Assumptions:
- The particle is described by a wave.
- The frequency of the wave is the energy E of the particle, while the momentum p is the wavenumber k.
- Failed to parse (Missing texvc executable;
please see math/README to configure.): E= \hbar \omega \;\;\;\;P= \hbar k \,
with Failed to parse (Missing texvc executable;
please see math/README to configure.): \scriptstyle \hbar
serving as a unit conversion factor.
- The total energy is the same function of momentum and position as in classical mechanics:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): E = {p^2\over 2m} + V(x)
- the first term is the kinetic energy and the second term is the potential energy.
Schrodinger required that a Wave packet at position x with wavenumber k will move along the trajectory determined by Newton's laws in the limit that the wavelength is small.
Consider first the case without a potential, V=0.
- Failed to parse (Missing texvc executable;
please see math/README to configure.): E = {1\over 2m} (p_x^2+p_y^2 + p_z^2)
Replacing energy/momentum by frequency/wavenumber:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \omega = {\hbar\over 2m} (k_x^2 + k_y^2 + k_z^2)
A plane wave is a wave with a wavenumber k, and it has the following form:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi = e^{i(k \cdot x- \omega t)}
The sign convention for the frequency is chosen to make the quantity in the exponential the relativistic dot-product.
Taking a time derivative multiplies by the frequency
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \frac{\partial}{\partial t} \psi = -i\omega \psi
Taking a spatial derivative multiplies by the wavenumber:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \frac{\partial}{\partial x} \psi = i k_x \psi\;\;\;\;\;\;{\partial^2 \over \partial x^2} \psi = -k_x^2 \psi
so that a plane wave with the right energy/frequency relationship obeys the free Schrodinger equation:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): i\hbar{\partial \over \partial t} \psi = -{\hbar^2 \over 2m} ( {\partial^2 \psi \over \partial x^2} + {\partial^2 \psi \over \partial y^2} + {\partial^2 \psi \over \partial z^2} )
Since every function is a linear combination of plane waves, this equation is obeyed by an arbitrary wave describing a free particle.
Since there is no potential, a wavepacket should travel in a straight line at the correct classical velocity. The velocity v of such a wavepacket is given by:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): v = {\partial \omega \over \partial k } = {\partial \over \partial k} {\hbar k^2\over 2m} = {\hbar k\over m}
which is the momentum over the velocity as it should be. This reproduces one of Hamilton's equations from mechanics:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): {dx \over dt} = {\partial H \over \partial p}
after identifying the energy and momentum of a wavepacket as the frequency and wavenumber.
To include a potential energy, consider that as a particle moves the energy is conserved, so that for a wavepacket with approximate wavenumber k at approximate position x the quantity
- Failed to parse (Missing texvc executable;
please see math/README to configure.): { \hbar^2 k^2\over 2m } + V(x)
must be constant. The frequency doesn't change as a wave moves, but the wavenumber does. So where there is a potential energy, it must add in the same way:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): i\hbar\frac{\partial}{\partial t}\psi=-\frac{\hbar^2}{2m}\nabla^2\psi + V(x)\psi
This is the time dependent schrodinger equation. It is the equation for the energy in classical mechanics, turned into a differential equation by substituting:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): E\rightarrow i\hbar{\partial\over \partial t} \;\;\;\;\;\; p\rightarrow -i\hbar{\partial\over \partial x}
Schrödinger studied the standing wave solutions, since these were the energy levels. Standing waves have a complicated dependence on space, but vary in time in a simple way:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi(x,t) = \psi(x) e^{-{iEt\over \hbar}}
substituting, the time-dependent equation becomes the standing wave equation:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): {E}\psi(x) = - {\hbar^2 \over 2m} \nabla^2 \psi(x) + V(x) \psi(x)
Which is the original time-independent Schrodinger equation.
In a potential gradient, the k-vector of a short-wavelength wave must vary from point to point,
to keep the total energy constant. Sheets perpendicular to the k-vector are the wavefronts,
and they gradually change direction, because the wavelength is not everywhere the same. A
wavepacket follows the shifting wavefronts with the classical velocity, with the acceleration equal
to the force divided by the mass.
an easy modern way to verify that Newton's second law holds for wavepackets is to take the Fourier transform of
the time dependent Schrodinger equation. For an arbitrary polynomial potential this is called the Schrodinger equation in the momentum representation:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): i\hbar {\partial \psi(p) \over \partial t} = {p^2\over 2m} \psi(p) + V(i\hbar{\partial\over \partial x}) \psi(p)
The group-velocity relation for the fourier trasformed wave-packet gives the second of Hamilton's equations.
- Failed to parse (Missing texvc executable;
please see math/README to configure.): {dp \over dt} = -{\partial H \over \partial x}
Shorter heuristic derivation
Assumptions:
- 1- The total energy E of a particle is
- Failed to parse (Missing texvc executable;
please see math/README to configure.): E=\frac{p^2}{2m}+V
- This is the classical mechanics expression for a particle with mass m where the total energy E is the sum of the kinetic energy, Failed to parse (Missing texvc executable;
please see math/README to configure.): \frac{p^2}{2m}
, and the potential energy V. The momentum of the particle is p, or mass times velocity. The potential energy is assumed to vary with position, and possibly time as well.
Note that the energy E and momentum p appear in the following two relations:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): E = h f\;
- where the frequency f of the quanta of radiation (photons) are related by Planck's constant h.
- Failed to parse (Missing texvc executable;
please see math/README to configure.): p=h / \lambda\;
- where Failed to parse (Missing texvc executable;
please see math/README to configure.): \lambda\,
is the wavelength of the wave. This hypothesis also requires:
- 4- The association of a wave (with wavefunction Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi
) with any particle.
Combining the above assumptions yields Schrödinger's equation:
Expressing frequency f in terms of angular frequency Failed to parse (Missing texvc executable;
please see math/README to configure.): \omega = 2\pi f\;
and wavelength Failed to parse (Missing texvc executable;
please see math/README to configure.): \lambda\,
in terms of wavenumber Failed to parse (Missing texvc executable;
please see math/README to configure.): k = 2\pi / \lambda\;
, with Failed to parse (Missing texvc executable;
please see math/README to configure.): \hbar = h / 2 \pi\;
we get:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): E=\hbar \omega
and
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \mathbf{p}=\hbar \mathbf{k}\;
where we have expressed p and k as vectors.
Schrödinger's great insight, late in 1925, was to express the phase of a plane wave as a complex phase factor:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi \approx e^{i(\mathbf{k}\cdot\mathbf{x}- \omega t)}
and to realize that since
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \frac{\partial}{\partial t} \psi = -i\omega \psi
then
- Failed to parse (Missing texvc executable;
please see math/README to configure.): E \psi = \hbar \omega \psi = i\hbar\frac{\partial}{\partial t} \psi
and similarly since:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \frac{\partial}{\partial x} \psi = i k_x \psi
then
- Failed to parse (Missing texvc executable;
please see math/README to configure.): p_x \psi = \hbar k_x \psi = -i\hbar\frac{\partial}{\partial x} \psi
and hence:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): p_x^2 \psi = -\hbar^2\frac{\partial^2}{\partial x^2} \psi
so that, again for a plane wave, he obtained:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): p^2 \psi = (p_x^2 + p_y^2 + p_z^2) \psi = -\hbar^2\left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\right) \psi = -\hbar^2\nabla^2 \psi
And by inserting these expressions for the energy and momentum into the classical mechanics formula we started with we get Schrödinger's famed equation for a single particle in the 3-dimensional case in the presence of a potential V:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): i\hbar\frac{\partial}{\partial t}\psi=-\frac{\hbar^2}{2m}\nabla^2\psi + V\psi
Using this equation, Schrödinger computed the spectral lines for hydrogen by treating a hydrogen atom's single negatively charged electron as a wave, Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi\;
, moving in a potential well, V, created by the positively charged proton. This computation reproduced the energy levels of the Bohr model.
But this was not enough, since Sommerfeld had already seemingly correctly reproduced relativistic corrections. Schrödinger used the relativistic energy momentum relation to find what is now known as the Klein-Gordon equation in a Coulomb potential (in natural units):
- Failed to parse (Missing texvc executable;
please see math/README to configure.): (E + {e^2\over r} )^2 \psi = - \nabla^2\psi + m^2 \psi
He found the standing-waves of this relativistic equation, but the relativistic corrections disagreed with Sommerfeld's formula. Discouraged, he put away his calculations and secluded himself in an isolated mountain cabin with a lover.
While there, Schrödinger decided that the earlier nonrelativistic calculations were novel enough to publish, and decided to leave off the problem of relativistic corrections for the future. He put together his wave equation and the spectral analysis of hydrogen in a paper in 1926.[2]. The paper was enthusiastically endorsed by Einstein, who saw the matter-waves as the visualizable antidote to what he considered to be the overly formal matrix mechanics.
The Schrödinger equation defines the behaviour of Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi\;
, but does not interpret what Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi\;
is. Schrödinger tried unsuccessfully to interpret it as a charge density.[citation needed] In 1926 Max Born, just a few days after Schrödinger's fourth and final paper was published, successfully interpreted Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi\;
as a probability amplitude[3]. Schrödinger, though, always opposed a statistical or probabilistic approach, with its associated discontinuities; like Einstein, who believed that quantum mechanics was a statistical approximation to an underlying deterministic theory, Schrödinger was never reconciled to the Copenhagen interpretation.[4]
[edit] Mathematical forms
There are various closely related equations which go under Schrödinger's name,
[edit] Time-dependent Schrödinger equation
The time-dependent Schrödinger equation for a system with energy operator Failed to parse (Missing texvc executable;
please see math/README to configure.): \hat H
is,
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \hat H \psi\left(\mathbf{r}, t\right) = i \hbar \frac{\partial \psi}{\partial t} \left(\mathbf{r}, t\right)
where Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi
is the wavefunction, Failed to parse (Missing texvc executable;
please see math/README to configure.): \hbar
is the reduced Planck's constant and Failed to parse (Missing texvc executable;
please see math/README to configure.): i
is the imaginary unit. The form of the Hamiltonian is different for different systems.
For a non-relativistic particle moving in a potential, the Hamiltonian operator is the sum of the kinetic and potential energies :
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \hat H = - \frac{\hbar^2}{2m} \nabla^2 + V\left(\mathbf{r}\right)
And the Schrödinger equation is a partial differential equation
- Failed to parse (Missing texvc executable;
please see math/README to configure.): -\frac{\hbar^2}{2 m} \left[\frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2} + \frac{\partial^2 \psi}{\partial z^2} \right] + V \psi = i \hbar \frac{\partial \psi}{\partial t}
[edit] Time-independent Schrödinger equation
When the Hamiltonian does not depend on time, for a single particle this is when the potential energy does not change in time, there are special solutions of the time-dependent equation which form standing waves. These waves have a constant energy/frequency, oscillating in time without changing shape. They obey what is sometimes called the time-independent Schrödinger equation
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \hat H \psi = E \psi \,
which means that
- Failed to parse (Missing texvc executable;
please see math/README to configure.): i \hbar \frac{\partial \psi}{\partial t} = E \psi
so that Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi
has constant frequency.
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi\left(\mathbf{r}, t\right) = \phi\left(\mathbf{r}\right) e^{-iEt/\hbar}
where Failed to parse (Missing texvc executable;
please see math/README to configure.): \phi\left(\mathbf{r}\right)
is the value of Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi
at Failed to parse (Missing texvc executable;
please see math/README to configure.): t=0
. Such a solution describes a stationary state in quantum mechanics, a state with a definite value of the energy. In such a state, all the probabilities for the outcomes of any measurement do not depend on time.
For a particle in a one-dimensional potential, the standing-wave condition is:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): -\frac{\hbar^2}{2 m} \frac{d^2 \phi (x)}{dx^2} + V(x) \phi (x) = E \phi (x).
In more dimensions, the only difference is more space derivatives:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \left[-\frac{\hbar^2}{2 m} \nabla^2 + V(\mathbf{r}) \right] \phi (\mathbf{r}) = E \phi (\mathbf{r}),
where Failed to parse (Missing texvc executable;
please see math/README to configure.): \nabla^2
is the Laplacian.
[edit] Bra-ket versions
In the mathematical formulation of quantum mechanics, a physical system is fully described by a vector in a complex Hilbert space, the collection of all possible normalizable wavefunctions. A normalizable wavefunction is just an alternate name for a vector in Hilbert space, the two terms are synonyms. This is true even though in general the vectors do not describe the probability amplitudes for a particle to be in a certain position, so they don't "wave" in any physical sense. The only wavefunction that is a wave in space and time is the wavefunction for a single particle in the position representation.
Two nonzero vectors which are multiples of each other, two wavefunctions which are the same up to rescaling, represent the same physical state. The Schrödinger equation is the rate of change of the state vector.
In bra-ket notation:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): i\hbar {d \over dt } |\psi\rangle = \hat H(t)|\psi\rangle
where Failed to parse (Missing texvc executable;
please see math/README to configure.): |\psi\rangle
is a ket, an infinite component complex vector. and Failed to parse (Missing texvc executable;
please see math/README to configure.): \hat H(t)
is the Hamiltonian, a linear map from kets to kets. The Hamiltonian should be a self-adjoint operator, so that its eigenvalues are real.
The nonzero elements of a Hilbert space are by definition normalizable and it is convenient to represent a state by an element of the ray which has unit length.
For every time-independent Hamiltonian operator, Failed to parse (Missing texvc executable;
please see math/README to configure.): \hat H
, there exists a set of quantum states, Failed to parse (Missing texvc executable;
please see math/README to configure.): \left|\psi_n\right\rang
, known as energy eigenstates, and corresponding real numbers Failed to parse (Missing texvc executable;
please see math/README to configure.): E_n
satisfying the eigenvalue equation,
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \hat H \left|\psi_n\right\rang = E_n \left|\psi_n \right\rang.
Alternatively, Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi
is said to be an eigenstate (eigenket) of Failed to parse (Missing texvc executable;
please see math/README to configure.): \hat H
with eigenvalue Failed to parse (Missing texvc executable;
please see math/README to configure.): E
. Such a state possesses a definite total energy, whose value Failed to parse (Missing texvc executable;
please see math/README to configure.): E_n
is the eigenvalue of the Hamiltonian. The corresponding eigenvector Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi_n\,
is normalizable to unity. This eigenvalue equation is referred to as the time-independent Schrödinger equation.
We purposely left out the variable(s) on which the wavefunction Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi_n\,
depends.
Self-adjoint operators, such as the Hamiltonian, have the property that their eigenvalues are always real numbers, as we would expect, since the energy is a physically observable quantity. Sometimes more than one linearly independent state vector correspond to the same energy Failed to parse (Missing texvc executable;
please see math/README to configure.): E_n
. If the maximum number of linearly independent eigenvectors corresponding to Failed to parse (Missing texvc executable;
please see math/README to configure.): E_n
equals k, we say that the energy level Failed to parse (Missing texvc executable;
please see math/README to configure.): E_n
is k-fold degenerate. When k=1 the energy level is called non-degenerate.
On inserting a solution of the time-independent Schrödinger equation into the full Schrödinger equation, we get
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \mathrm{i} \hbar \frac{\partial}{\partial t} \left| \psi_n \left(t\right) \right\rangle = E_n \left|\psi_n\left(t\right)\right\rang.
It is relatively easy to solve this equation. One finds that the energy eigenstates
(i.e., solutions of the time-independent Schrödinger equation) change as a function of time only trivially, namely, only by a complex phase:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \left| \psi \left(t\right) \right\rangle = \mathrm{e}^{-\mathrm{i} Et / \hbar} \left|\psi\left(0\right)\right\rang.
It immediately follows that the probability amplitude,
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi(t)^*\psi(t) = \mathrm{e}^{\mathrm{i} Et / \hbar}\mathrm{e}^{-\mathrm{i} Et / \hbar} \psi(0)^*\psi(0) = |\psi(0)|^2,
is time-independent. Because of a similar cancellation of phase factors in bra and ket, all average (expectation) values of time-independent observables (physical quantities) computed from Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi(t)\,
are time-independent.
Energy eigenstates are convenient to work with because they form a complete set of states. That is, the eigenvectors Failed to parse (Missing texvc executable;
please see math/README to configure.): \left\{\left|n\right\rang\right\}
form a basis for the state space. We introduced here the short-hand notation
Failed to parse (Missing texvc executable;
please see math/README to configure.): |\,n\,\rang = \psi_n
.
Then any state vector that is a solution of the time-dependent Schrödinger equation (with a time-independent Failed to parse (Missing texvc executable;
please see math/README to configure.): \hat H
)
Failed to parse (Missing texvc executable;
please see math/README to configure.): \left|\psi\left(t\right)\right\rang
can be written as a linear superposition of energy eigenstates:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \left|\psi\left(t\right)\right\rang = \sum_n c_n(t) \left|n\right\rang \quad,\quad \hat H \left|n\right\rang = E_n \left|n\right\rang \quad,\quad \sum_n \left|c_n\left(t\right)\right|^2 = 1.
(The last equation enforces the requirement that Failed to parse (Missing texvc executable;
please see math/README to configure.): \left|\psi\left(t\right)\right\rang
,
like all state vectors, may be normalized to a unit vector.) Applying the Hamiltonian operator to each side of the first equation, the time-dependent Schrödinger equation in the left-hand side and using the fact that the energy basis vectors are by definition linearly independent, we readily obtain
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \mathrm{i}\hbar \frac{\partial c_n}{\partial t} = E_n c_n\left(t\right).
Therefore, if we know the decomposition of Failed to parse (Missing texvc executable;
please see math/README to configure.): \left|\psi\left(t\right)\right\rang
into the energy basis at time Failed to parse (Missing texvc executable;
please see math/README to configure.): t = 0
, its value at any subsequent time is given simply by
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \left|\psi\left(t\right)\right\rang = \sum_n \mathrm{e}^{-\mathrm{i}E_nt/\hbar} c_n\left(0\right) \left|n\right\rang.
Note that when some values Failed to parse (Missing texvc executable;
please see math/README to configure.): c_n(0)\,
are not equal to zero for
differing energy values Failed to parse (Missing texvc executable;
please see math/README to configure.): E_n\,
, the left-hand side is not an eigenvector of
the energy operator Failed to parse (Missing texvc executable;
please see math/README to configure.): \hat H
. The left-hand is an eigenvector when
the only Failed to parse (Missing texvc executable;
please see math/README to configure.): c_n(0)\,
-values not equal to zero belong the same energy, so that
Failed to parse (Missing texvc executable;
please see math/README to configure.): \mathrm{e}^{-\mathrm{i}E_nt/\hbar}
can be factored out. In many real-world
application this is the case and the state vector Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi(t)\,
(containing time only in its phase factor) is then a solution of the time-independent Schrödinger equation.
Example
Let Failed to parse (Missing texvc executable;
please see math/README to configure.): |\,1\,\rangle
and Failed to parse (Missing texvc executable;
please see math/README to configure.): |\,2\,\rangle
be degenerate eigenstates of the time-independent Hamiltonian Failed to parse (Missing texvc executable;
please see math/README to configure.): \hat H\,
-
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \hat H\,|\,1\,\rangle = E |\,1\,\rangle \quad \hbox{and} \quad \hat H\,|\,2\,\rangle = E |\,2\,\rangle.
Suppose a solution Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi(t)\,
of the full (time-dependent) Schrödinger equation of Failed to parse (Missing texvc executable;
please see math/README to configure.): \hat H\,
has the form at t = 0:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): |\,\psi(0)\,\rangle = c_1 |\,1\,\rangle + c_2 |\,2\,\rangle.
Hence, because of the discussion above, at t > 0 :
- Failed to parse (Missing texvc executable;
please see math/README to configure.): |\,\psi(t)\,\rangle = \mathrm{e}^{-\mathrm{i}Et/\hbar} c_1 |\,1\,\rangle + \mathrm{e}^{-\mathrm{i}Et/\hbar} c_2 |\,2\,\rangle = \mathrm{e}^{-\mathrm{i}Et/\hbar} \left( c_1 |\,1\,\rangle + c_2 |\,2\,\rangle\right) = \mathrm{e}^{-\mathrm{i}Et/\hbar}|\,\psi(0)\,\rangle,
which shows that Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi(t)\,
only depends on time in a trivial way (through its phase),
also in the case of degeneracy.
Apply now Failed to parse (Missing texvc executable;
please see math/README to configure.): \hat H\,
-
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \hat H\,|\,\psi(t)\,\rangle = \mathrm{e}^{-\mathrm{i}Et/\hbar} c_1 E\,|\,1\,\rangle + \mathrm{e}^{-\mathrm{i}Et/\hbar} c_2 E\, |\,2\,\rangle = E\mathrm{e}^{-\mathrm{i}Et/\hbar} \left( c_1 |\,1\,\rangle + c_2 |\,2\,\rangle\right)
- Failed to parse (Missing texvc executable;
please see math/README to configure.): = E \mathrm{e}^{-\mathrm{i}Et/\hbar}|\,\psi(0)\,\rangle = E\,|\,\psi(t)\,\rangle.
Conclusion: The wavefunction Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi(t)\,
with the given initial condition (its form at t = 0), remains a solution of the time-independent Schrödinger equation
Failed to parse (Missing texvc executable;
please see math/README to configure.): \hat H\psi(t) = E\psi(t)
for all times t > 0.
[edit] Properties
[edit] Linearity
The Schrödinger equation (in any form) is linear in the wavefunction, meaning that if Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi(x, t)
and Failed to parse (Missing texvc executable;
please see math/README to configure.): \phi(x,t)
are solutions, then so is Failed to parse (Missing texvc executable;
please see math/README to configure.): a \psi + b \phi
, where a and b are any complex numbers. This property of the Schrödinger equation has important consequences.
Linearity of the Schrödinger equation
Assumptions:
- The Schrödinger equation:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \hat H(t)\left|\psi(t)\right\rangle = i\hbar \frac{d}{d t} \left| \psi(t) \right\rangle
- Failed to parse (Missing texvc executable;
please see math/README to configure.): |\psi\rangle
and Failed to parse (Missing texvc executable;
please see math/README to configure.): |\phi\rangle
are solutions of the Schrödinger equation.
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \hat H(t)\left|a\psi(t) + b\phi(t)\right\rangle
- Failed to parse (Missing texvc executable;
please see math/README to configure.): =\hat H(t)\left|a\psi(t)\right\rangle + \hat H(t)\left|b\phi(t)\right\rangle
(as the Hamiltonian is a linear operator)
- Failed to parse (Missing texvc executable;
please see math/README to configure.): = a \left( \hat H(t)\left|\psi(t)\right\rangle \right) + b \left( \hat H(t)\left|\phi(t)\right\rangle \right)
- Failed to parse (Missing texvc executable;
please see math/README to configure.): =a \left( i\hbar \frac{d}{d t} \left| \psi(t) \right\rangle \right) + b \left( i\hbar \frac{d}{d t} \left| \phi(t) \right\rangle \right)
- Failed to parse (Missing texvc executable;
please see math/README to configure.): = i \hbar \left( \frac{d}{d t} \left| a\psi(t) \right\rangle \right) + i \hbar \left( \frac{d}{d t} \left| b\phi(t) \right\rangle \right)
- Failed to parse (Missing texvc executable;
please see math/README to configure.): = i \hbar \frac{d}{d t} \left| a\psi(t) + b\phi(t) \right\rangle
[edit] Conservation of probability
In order to describe how probability density changes with time, we define a probability current or probability flux. The probability flux represents a flowing of probability across space.
For example, consider a Gaussian probability curve centered around Failed to parse (Missing texvc executable;
please see math/README to configure.): x_0
with Failed to parse (Missing texvc executable;
please see math/README to configure.): x_0
moving at speed Failed to parse (Missing texvc executable;
please see math/README to configure.): v
to the right. One may say that the probability is flowing towards the right, i.e., there is a probability flux directed to the right.
The probability flux Failed to parse (Missing texvc executable;
please see math/README to configure.): \mathbf{j}
is defined as:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \mathbf{j} = {\hbar \over m} \cdot {1 \over {2 \mathrm{i}}} \left( \psi ^{*} \nabla \psi - \psi \nabla \psi^{*} \right) = {\hbar \over m} \operatorname{Im} \left( \psi ^{*} \nabla \psi \right)
and measured in units of (probability)/(area × time) = r−2t−1.
The probability flux satisfies the required continuity equation for a conserved quantity, i.e.:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): { \partial \over \partial t} P\left(x,t\right) + \nabla \cdot \mathbf{j} = 0
where Failed to parse (Missing texvc executable;
please see math/README to configure.): P\left(x, t\right)
is the probability density and measured in units of (probability)/(volume) = r−3.
This equation is the mathematical equivalent of the probability conservation law.
A standard calculation shows that for a plane wave described by the wavefunction,
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi (x,t) = \, A e^{ \mathrm{i} (k x - \omega t)}
the probability flux is given by
- Failed to parse (Missing texvc executable;
please see math/README to configure.): j\left(x,t\right) = \left|A\right|^2 {k \hbar \over m}
showing that not only is the probability of finding the particle in a plane wave state the same everywhere at all times, but also that it is moving at constant speed everywhere.
[edit] Correspondence principle
-
The Schrödinger equation satisfies the correspondence principle.
[edit] Solutions
-
Analytical solutions of the time-independent Schrödinger equation can be obtained for a variety of relatively simple conditions. These solutions provide insight into the nature of quantum phenomena and sometimes provide a reasonable approximation of the behavior of more complex systems (e.g., in statistical mechanics, molecular vibrations are often approximated as harmonic oscillators). Several of the more common analytical solutions can be found in the list of quantum mechanical systems with analytical solutions.
For many systems, however, there is no analytic solution to the Schrödinger equation. In these cases, one must resort to approximate solutions. Some of the common techniques are:
[edit] Free Schrödinger equation
When the potential is zero, the Schrödinger equation is linear with constant coefficients:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): i \frac{\partial \psi}{\partial t}=-{1\over 2m}\nabla^2\psi
where Failed to parse (Missing texvc executable;
please see math/README to configure.): \scriptstyle \hbar
has been set to 1. The solution Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi_t(x)
for any initial condition Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi_0(x)
can be found by Fourier transforms. Because the coefficients are constant, an initial plane wave:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi_0(x) = A e^{i k x} \,
stays a plane wave. Only the coefficient changes. Substituting:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): {dA \over dt} = -{i k^2 \over 2m} A \,
So that A is also oscillating in time:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): A(t) = A e^{- i {k^2 \over 2m} t} \,
and the solution is:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi_t(x) = A e^{i k x - i \omega t} \,
Where Failed to parse (Missing texvc executable;
please see math/README to configure.): \omega=k^2/2m
, a restatement of DeBroglie's relations.
To find the general solution, write the initial condition as a sum of plane waves by taking its Fourier transform:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi_0(x) = \int_k \psi(k) e^{ikx} \,
The equation is linear, so each plane waves evolves independently:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi_t(x) = \int_k \psi(k)e^{-i\omega t} e^{ikx} \,
Which is the general solution. When complemented by an effective method for taking Fourier transforms, it becomes an efficient algorithm for finding the wavefunction at any future time--- Fourier transform the initial conditions, multiply by a phase, and transform back.
[edit] Gaussian Wavepacket
An easy and instructive example is the Gaussian wavepacket:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi_0(x) = e^{-x^2 / 2a} \,
where a is a positive real number, the square of the width of the wavepacket. The total normalization of this wavefunction is:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \langle \psi|\psi\rangle = \int_x \psi^* \psi = \sqrt{\pi a}
The Fourier transform is a Gaussian again in terms of the wavenumber k:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi_0(k) = (2\pi a)^{d/2} e^{- a k^2/2} \,
With the physics convention which puts the factors of Failed to parse (Missing texvc executable;
please see math/README to configure.): 2\pi
in Fourier transforms in the k-measure.
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi_0(x) = \int_k \psi_0(k) e^{-ikx} = \int {d^dk \over (2\pi)^d} \psi_0(k) e^{-ikx}
Each separate wave only phase-rotates in time, so that the time dependent Fourier-transformed solution is:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi_t(k) = (2\pi a)^{d/2} e^{- a { k^2\over 2} - it {k^2\over 2m}} = (2\pi a)^{d/2} e^{-(a+it/m){k^2\over 2}} \,
The inverse Fourier transform is still a Gaussian, but the parameter a has become complex, and there is an overall normalization factor.
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi_t(x) = \left({a \over a + i t/m}\right)^{d/2} e^{ {-x^2\over 2(a + i t/m)} } \,
The sign of the square root is determined by continuity in time--- it is the value which is nearest to the positive square root of a. Since the mass only contributes to rescale the time, it is convenient at this point to rescale time to absorb m: Failed to parse (Missing texvc executable;
please see math/README to configure.): t\rightarrow t/m
The integral of Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi
over all space is invariant, because it is the inner product of Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi
with the state of lowest energy, which is a wave with infinite wavelength, a constant function of space. This is generally true: for any energy state, with wavefunction Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi_0
, the inner product:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \langle \psi_0 | \psi \rangle = \int_x \psi_0(x) \psi_(x)
,
is constant in time. This provides a check on the normalization factor.
The sum of the absolute square of Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi
is also invariant, which is a statement of the conservation of probability. Explicitly in one dimension:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): |\psi|^2 = \psi\psi^* = {a \over \sqrt{a^2+t^2} } e^{-{x^2 a \over a^2 + t^2}}
Which gives the norm:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \int |\psi|^2 = \sqrt{\pi a}
which has preserved its value, as it must.
The width of the Gaussian is the interesting quantity, and it can be read off from the form of Failed to parse (Missing texvc executable;
please see math/README to configure.): |\psi^2|
-
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \sqrt{a^2 + t^2 \over a} \,
.
The width eventually grows linearly in time, as Failed to parse (Missing texvc executable;
please see math/README to configure.): \scriptstyle t/\sqrt{a}
. This is called wave-packet spreading--- no matter how narrow the initial wavefunction, a Schrodinger wave eventually fills all of space. The linear growth is a reflection of the momentum uncertainty--- the wavepacket is confined to a narrow width Failed to parse (Missing texvc executable;
please see math/README to configure.): \scriptstyle \sqrt{a}
and so has a momentum which is uncertain by the reciprocal amount Failed to parse (Missing texvc executable;
please see math/README to configure.): \scriptstyle 1/\sqrt{a}
, a spread in velocity of Failed to parse (Missing texvc executable;
please see math/README to configure.): 1/m\sqrt{a}
, and therefore in the future position by Failed to parse (Missing texvc executable;
please see math/README to configure.): \scriptstyle t/m\sqrt{a}
, where the factor of m has been restored by undoing the earlier rescaling of time.
A more general form of the same solution can be found by applying a Galilean boost. This replaces x by x-vt and changes the phase of the wavefunction by Failed to parse (Missing texvc executable;
please see math/README to configure.): \scriptstyle e^{-imvx}
. For the initial condition (normalized to absorb the numerator of the final wavefunction):
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi_0 = {1\over \sqrt{2\pi a}} e^{{ x^2\over 2 a } + i v x} \,
The time dependent solution is the boosted phase-rotated version of the previous wavepacket:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi_t(x) = {1\over \sqrt{2\pi(a+i t)}} e^{{(x-vt)^2 \over 2 (a^2 +it)} + i v x } \,
Which moves with a steady velocity, but otherwise spreads in the same way.
[edit] Free Propagator
The narrow-width limit of the Gaussian wavepacket solution is the propagator K. For other differential equations, this is sometimes called the Green's function, but in quantum mechanics it is traditional to reserve the name Green's function for the time Fourier transform of K. When a is the infinitesimal quantity Failed to parse (Missing texvc executable;
please see math/README to configure.): \epsilon
, the Gaussian initial condition, rescaled so that its integral is one:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi_0(x) = {1\over \sqrt{2\pi \epsilon} } e^{-{x^2\over 2\epsilon}} \,
becomes a delta function, so that its time evolution:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): K_t(x) = {1\over \sqrt{2\pi (i t + \epsilon)}} e^{ - x^2 \over 2it+\epsilon } \,
gives the propagator.
Note that a very narrow initial wavepacket instantly becomes infinitely wide, with a phase which is more rapidly oscillatory at large values of x. This might seem strange--- the solution goes from being concentrated at one point to being everywhere at all later times, but it is a reflection of the momentum uncertainty of a localized particle. Also note that the norm of the wavefunction is infinite, but this is also correct since the square of a delta function is divergent in the same way.
The factor of Failed to parse (Missing texvc executable;
please see math/README to configure.): \epsilon
is an infinitesimal quantity which is there to make sure that integrals over K are well defined. In the limit that Failed to parse (Missing texvc executable;
please see math/README to configure.): \epsilon
becomes zero, K becomes purely oscillatory and integrals of K are not absolutely convergent. In the remainder of this section, it will be set to zero, but in order for all the integrations over intermediate states to be well defined, the limit Failed to parse (Missing texvc executable;
please see math/README to configure.): \scriptstyle \epsilon\rightarrow 0
is to only to be taken after the final state is calculated.
The propagator is the amplitude for reaching point x at time t, when starting at the origin, x=0. By translation invariance, the amplitude for reaching a point x when starting at point y is the same function, only translated:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): K_t(x,y) = K_t(x-y) = {1\over \sqrt{2\pi it}} e^{-i(x-y)^2 \over 2t} \,
In the limit when t is small, the propagator converges to a delta function:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \lim_{t\rightarrow 0} K_t(x-y) = \delta(x-y)
but only in the sense of distributions. The integral of this quantity multiplied by an arbitrary differentiable test function gives the value of the test function at zero. To see this, note that the integral over all space of K is equal to 1 at all times:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \int_x K_t(x) = 1 \,
since this integral is the inner-product of K with the uniform wavefunction. But the phase factor in the exponent has a nonzero spatial derivative everywhere except at the origin, and so when the time is small there are fast phase cancellations at all but one point. This is rigorously true when the limit Failed to parse (Missing texvc executable;
please see math/README to configure.): \epsilon\rightarrow zero
is taken after everything else.
So the propagation kernel is the future time evolution of a delta function, and it is continuous in a sense, it converges to the initial delta function at small times. If the initial wavefunction is an infinitely narrow spike at position Failed to parse (Missing texvc executable;
please see math/README to configure.): x_0
-
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi_0(x) = \delta(x - x_0) \,
it becomes the oscillatory wave:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi_t(x) = {1\over \sqrt{2\pi i t}} e^{ -i (x-x_0) ^2 /2t} \,
Since every function can be written as a sum of narrow spikes:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi_0(x) = \int_y \psi_0(y) \delta(x-y) \,
the time evolution of every function is determined by the propagation kernel:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi_t(x) = \int_y \psi_0(x) {1\over \sqrt{2\pi it}} e^{-i (x-x_0)^2 / 2t} \,
And this is an alternate way to express the general solution. The intepretation of this expression is that the amplitude for a particle to be found at point x at time t is the amplitude that it started at Failed to parse (Missing texvc executable;
please see math/README to configure.): x_0
times the amplitude that it went from Failed to parse (Missing texvc executable;
please see math/README to configure.): x_0
to x, summed over all the possible starting points. In other words, it is a convolution of the kernel K with the initial condition.
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi_t = K * \psi_0 \,
Since the amplitude to travel from x to y after a time Failed to parse (Missing texvc executable;
please see math/README to configure.): t+t'
can be considered in two steps, the propagator obeys the identity:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \int_y K(x-y;t)K(y-z;t') = K(x-z;t+t') \,
Which can be interpreted as follows: the amplitude to travel from x to z in time t+t' is the sum of the amplitude to travel from x to y in time t multiplied by the amplitude to travel from y to z in time t', summed over all possible intermediate states y. This is a property of an arbitrary quantum system, and by subdividing the time into many segments, it allows the time evolution to be expressed as a path integral.
[edit] Analytic Continuation to Diffusion
The spreading of wavepackets in quantum mechanics is directly related to the spreading of probability densities in diffusion. For a particle which is random walking, the probability density function at any point satisfies the diffusion equation:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): {\partial \over \partial t} \rho = {1\over 2} {\partial \over \partial x^2 } \rho
where the factor of 2, which can be removed by a rescaling either time or space, is only for convenience.
A solution of this equation is the spreading gaussian:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \rho_t(x) = {1\over \sqrt{2\pi t}} e^{-x^2 \over 2t}
and since the integral of Failed to parse (Missing texvc executable;
please see math/README to configure.): \rho_t
, is constant, while the width is becoming narrow at small times, this function approaches a delta function at t=0:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \lim_{t\rightarrow 0} \rho_t(x) = \delta(x) \,
again, only in the sense of distributions, so that
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \lim_{t\rightarrow 0} \int_x f(x) \rho_t(x) = f(0) \,
for any smooth test function f.
The spreading Gaussian is the propagation kernel for the diffusion equation, and it obeys the identity:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): K_{t+t'}(x) = K_{t}*K_{t'} \,
Which allows diffusion to be expressed as a path integral. The propagation is the exponential of an operator H:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): K_t(x) = e^{-tH} \,
which is the infinitesimal diffusion operator.
- Failed to parse (Missing texvc executable;
please see math/README to configure.): H= -{\nabla^2\over 2} \,
The exponential can be defined over a range of t's which include complex values, so long as integrals over the propagation kernel stay convergent.
- Failed to parse (Missing texvc executable;
please see math/README to configure.): K_t(x) = e^{-zH} \,
As long as the real part of z is positive, for large values of x K is exponentially decreasing and integrals over K are absolutely convergent.
The limit of this expression for z coming close to the pure imaginary axis is the Schrodinger propagator:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): K_t(x) = e^{-(iz+\epsilon)H} \,
and this gives a more conceptual explanation for the time evolution of Gaussians. From the fundamental identity of exponentiation, or path integration:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): K_z * K_{z'} = K_{z+z'} \,
Holds for all complex z values where the integrals are absolutely convergent so that the operators are well defined.
So that quantum evolution starting from a Gaussian:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi_0 = K_t(x) \,
gives the time evolved state:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi_t = K_{it} * K_t = K_{t+it} \,
This explains the diffusive form of the Gaussian solutions:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi_t(x) = {1\over \sqrt{a+it} } e^{- {x^2\over 2(a+it)} } \,
[edit] Variational Principle
The variational principle asserts that for any any Hermitian matrix A, the lowest eigenvalue minimizes the quantity:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \langle v,Av \rangle = \sum_{ij} A_{ij} v^*_i v_j \,
on the unit sphere Failed to parse (Missing texvc executable;
please see math/README to configure.): <v,v>=1
. This follows by the method of Lagrange multipliers, at the minimum the gradient of the function is parallel to the gradient of the constraint:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): {\partial\over \partial v_i} \langle v,Av\rangle = \lambda {\partial \over \partial v_i} \langle v,v\rangle \,
which is the eigenvalue condition
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \sum_{j} A_{ij} v_j = \lambda v_i \,
so that the extreme values of a quadratic form A are the eigenvalues of A, and the value of the function at the extreme values is just the corresponding eigenvalue:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \langle v,Av\rangle = \lambda\langle v,v\rangle = \lambda \,
When the hermitian matrix is the Hamiltonian, the minimum value is the lowest energy level.
In the space of all wavefunctions, the unit sphere is the space of all normalized wavefunctions Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi
, the ground state minimizes
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \langle \psi | H |\psi \rangle = \int \psi^* H \psi = \int \psi^* (-\nabla^2 + V(x)) \psi \,
or, after an integration by parts,
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \langle \psi | H |\psi \rangle = \int |\nabla \psi|^2 + V(x) |\psi|^2 \,
All the stationary points are real, since the integrand is real. In general, when a wavefunction Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi
is an eigenstate of the Schrodinger equation in a potential, the real and imaginary part of psi are both separately eigenstates with the same eigenvalue.
The lowest energy state has a positive definite wavefunction, because given a Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi
which minimizes the integral, Failed to parse (Missing texvc executable;
please see math/README to configure.): |\psi|
, the absolute value, is also a minimizer. But this minimizer has sharp corners at places where Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi
changes sign, and these sharp corners can be rounded out to reduce the gradient contribution.
[edit] Potential and Ground State
For a particle in a positive definite potential, the ground state wavefunction is real and positive, and has a dual interpretation as the probability density for a diffusion process. The analogy between diffusion and nonrelativistic quantum motion, originally discovered and exploited by Schrodinger, has led to many exact solutions.
A positive definite wavefunction:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi = e^{-W(x)} \,
is a solution to the time-independent Schrodinger equation with m=1 and potential:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): V(x) = {1\over 2} |\nabla W|^2 - {1\over 2} \nabla^2 W \,
with zero total energy. W, the logarithm of the ground state wavefunction. The second derivative term is higher order in Failed to parse (Missing texvc executable;
please see math/README to configure.): \scriptstyle \hbar
, and ignoring it gives the semiclassical approximation.
The form of the ground state wavefunction is motivated by the observation that the ground state wavefunction is the Boltzmann probability for a different problem, the probability for finding a particle diffusing in space with the free-energy at different points given by W. If the diffusion obeys detailed balance and the diffusion constant is everywhere the same, the Fokker Planck equation for this diffusion is the Schrodinger equation when the time parameter is allowed to be imaginary. This analytic continuation gives the eigenstates a dual interpretation--- either as the energy levels of a quantum system, or the relaxation times for a stochastic equation.
[edit] Harmonic Oscillator
W should grow at infinity, so that the wavefunction has a finite integral. The simplest analytic form is:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): W(x) = \omega x^2 \,
with an arbitrary constant Failed to parse (Missing texvc executable;
please see math/README to configure.): \omega
, which gives the potential:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): V(x) = {1\over 2} \omega^2 x^2 - {\omega \over 2} \,
This potential describes a Harmonic oscillator, with the ground state wavefunction:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi(x) = e^{-\omega x^2 } \,
The total energy is zero, but the potential is shifted by a constant. The ground state energy of the usual unshifted Harmonic oscillator potential:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): V(x) = {\omega x^2 \over 2} \,
is then the additive constant:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): E_0 = {a\over 2} \,
which is the zero point energy of the oscillator.
[edit] Coulomb Potential
Another simple but useful form is
- Failed to parse (Missing texvc executable;
please see math/README to configure.): W(x) = 2a|x| \,
where W is proportional to the radial coordinate. This is the ground state for two different potentials, depending on the dimension. In one dimension, the corresponding potential is singular at the origin, where it has some nonzero density:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): V(x) = 2a^2 + a \delta(x) \,
and, up to some rescaling of variables, this is the lowest energy state for a delta function potential, with the bound state energy added on.
- Failed to parse (Missing texvc executable;
please see math/README to configure.): V(x) = a \delta(x) \,
with the ground state energy:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): E_0 = - 2a^2 \,
and the ground state wavefunction:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi = e^{-2a|x|} \,
In higher dimensions, the same form gives the potential:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): V(x) = 2a^2+ { 2a (d-1) \over r}; \,
which can be identified as the attractive Coulomb law, up to an additive constant which is the ground state energy. This is the superpotential that describes the lowest energy level of the Hydrogen atom, once the mass is restored by dimensional analysis:
- Failed to parse (Missing texvc executable;
please see math/README to configure.): \psi_0 = e^{-r/r_0} \,
where Failed to parse (Missing texvc executable;
please see math/README to configure.): r_0
is the Bohr radius, with energy
- Failed to parse (Missing texvc executable;
please see math/README to configure.): E_0 = - {2a\over d-1} \,
The ansatz
- Failed to parse (Missing texvc executable;
please see math/README to configure.): W(x) = a r + b \log(r) \,
modifies the Coulomb potential to include a quadratic term proportional to Failed to parse (Missing texvc executable;
please see math/README to configure.): 1/r^2
, which is useful for nonzero angular momentum.
[edit] Relativistic generalisations
-
The Schrödinger equation does not take into account relativistic effects, meaning that the Schrödinger equation is invariant under a Galilean transformation, but not under a Lorentz transformation.
Galilean invariance of the Schrödinger equation
Relativistically valid generalisations incorporating ideas from special relativity include the Klein-Gordon equation and the Dirac equation.
[edit] Applications
[edit] See also
[edit] References
- ^ Schrödinger, Erwin (December 1926). "An Undulatory Theory of the Mechanics of Atoms and Molecules" (PDF). Phys. Rev. 28 (6): 1049 - 1070.
- ^ Erwin Schrödinger, Annalen der Physik, (Leipzig) (1926), Main paper
- ^ Schrödinger: Life and Thought by Walter John Moore, Cambridge University Press 1992 ISBN 0-521-43767-9, page 220
- ^ Schrödinger: Life and Thought by Walter John Moore, Cambridge University Press 1992 ISBN 0-521-43767-9, page 479 (hardback version) makes it clear that even in his last year of life, in a letter to Max Born, he never accepted the Copenhagen Interpretation. cf pg 220
[edit] Modern reviews
- David J. Griffiths (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 013805326X.
[edit] External links
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