Schrödinger equation

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The Schrödinger equation describes the time dependence of states in the Schrödinger picture. The time-dependent Schrödinger equation applies to all systems, while the the time-independent Schrödinger equation applies to systems for which the energy operator or Hamiltonian H\, does not depend explicitly on time.

Contents

Schrödinger's original formulation

Schrödinger attempted to make use of the de Broglie relations

\mathbf{p} = \hbar \mathbf{k},
E = \hbar \omega,

in order to develop a description of what was at the time called matter waves. Hypothesizing traveling plane waves of the form

\psi(\mathbf{r},t) = e^{i ( \mathbf{k} \cdot \mathbf{r} - \omega t)}\,,

he expressed the momentum \mathbf{p}\, and energy E\, as operators acting on the wave function \psi(\mathbf{r},t)\,, as

E \psi(\mathbf{r},t) = i \hbar \frac{\partial}{\partial t} \psi(\mathbf{r},t)\,,

and

\mathbf{p} \psi(\mathbf{r},t) = -i \hbar \boldsymbol{\nabla} \psi(\mathbf{r},t)\,.

Inserting the above relations into the relativistic expression for energy,

E^2 = p^2 c^2 + m^2 c^4\,,

he arrived at what is now known as the Klein-Gordon equation in 1925. However, the equation was difficult to interpret, and without knowing about spin, he obtained incorrect predictions of the fine structure of the Hydrogen atom. In January, 1926, Schrödinger published a non-relativistic approximation of the equation, using

E \approx \frac{p^2}{2m} + V(\mathbf{r})\,,

obtaining

(1)
i \hbar \frac{\partial}{\partial t} \psi(\mathbf{r},t) = \left[ -\frac{\hbar^2}{2m} \nabla^2 + V(x) \right] \psi(\mathbf{r}, t)\,,

where the potential energy V(x)\, has been introduced in a logical way. The term

H = \frac{p^2}{2m} + V(\mathbf{r})\,,

is simply the Hamiltonian from classical mechanics, and becomes the Hamiltonian operator

H = -\frac{\hbar^2}{2m} \nabla^2 + V(x)\,,

of the quantum description. More generally, the time-dependent Schrödinger equation in the position representation reads

i \hbar \frac{\partial}{\partial t} \psi(\mathbf{r},t) = H \psi(\mathbf{r},t)\,.

In many cases H\, does not explicitly depend on time, and using separation of variables, we may write

\psi(\mathbf{r},t) = A(t) \psi(\mathbf{r})\,,

where i \hbar \frac{\partial}{\partial t} A(t) = E A(t)\,, and where \psi(\mathbf{r})\, solves the time-independent Schrödinger equation

H \psi(\mathbf{r}) = E \psi(\mathbf{r})\,,

or

\left( -\frac{\hbar^2}{2m} \nabla^2 + V(x) \right)\psi(\mathbf{r}) = E \psi(\mathbf{r})\,.

Provided a solution to the first equation can be found, namely

A(t) = e^{-i E/\hbar t}\,,

one solution to the time-dependent equation is therefore

\psi(\mathbf{r},t) = e^{-i E/\hbar t} \psi(\mathbf{r})\,.

Such a solution is termed a stationary solution and has definite energy. The general solution is then a linear combination of such stationary solutions, consistent with boundary conditions.

Time-dependent Schrödinger equation

The time dependence of a state \left|\Psi(t)\right\rangle\, is given by

i \hbar \frac{\partial}{\partial t} \left| \Psi(t) \right\rangle = H(t) \left| \Psi(t) \right\rangle\,,

where H(t)\, is the (possibly time dependent) Hamiltonian of the system. This is the most generally version of the Schrödinger equation.

In terms of the time evolution operator U(t)\equiv U(t,0)\,, we can write

\left|\Psi(t)\right\rangle = U(t) \left|\Psi(0)\right\rangle\,,

so that

\frac{\partial}{\partial t} \left| \Psi(t) \right\rangle = \frac{\partial U(t)}{\partial t}\left| \Psi(t) \right\rangle\,.

These equations must be true for all states, and so

\frac{\partial U(t)}{\partial t} = \frac{1}{i\hbar} H(t)\,.

We will postpone the argument that time evolution is generated by the Hamiltonian until the discussion of Poisson brackets and the Heisenberg picture. For now, let us reason that it is a suitable generalization of the original Schrödinger equation.

In many cases, H\, does not depend explicitly on time, and this equation can be integrated:

U(t) = e^{-\frac{i}{\hbar} H t}\,,

so that

\left|\Psi(t)\right\rangle = e^{-\frac{i}{\hbar} H (t-t_0)} \left|\Psi(t_0)\right\rangle\,.

Because H\, is Hermitian, it has real eigenvalues, and a complete basis of eigenvectors \left\{\left|\phi_n\right\rangle\right\}\, with eigenvalues \left\{E_n\right\}\,. States that are eigenstates of H\, evolve according to their energy eigenvalues. Thus, if

H \left|\phi_n\right\rangle =E_n \left|\phi_n\right\rangle\,,

then

\left|\phi_n(t)\right\rangle = e^{-\frac{i}{\hbar} E_n (t-t_0)} \left|\phi_n(t_0)\right\rangle\,.

Furthermore, if \left|\Psi(0)\right\rangle\, is written in terms of these energy eigenstates, i.e., if

\left|\Psi(t_0)\right\rangle\ = \sum_n c_n \left|\phi_n(t_0) \right\rangle\,,

then

\left|\Psi(t)\right\rangle = \sum_n c_n e^{-\frac{i}{\hbar} E_n (t-t_0)} \left|\phi_n(t_0)\right\rangle\,.

For systems in which the Hamiltonian is time dependent, the time evolution operator may be written as

U(t) = \mathcal{T} e^{-\frac{i}{\hbar} \int_0^t \!dt'\, H(t') }\,,

where \mathcal{T}\, denotes the time-ordering of the operators in the expression.

Time-independent Schrödinger equation

Suppose the Hamiltonian does not have explicit time dependence, so that the system has time translation invariance. Suppose that at time t_0=0\, the system is in the state \left|\phi_n\right\rangle\,. Then

i \hbar \frac{\partial}{\partial t} \left| \Psi(0) \right\rangle = H \left| \phi_n \right\rangle = E_n \left|\phi_n\right\rangle\,.

The eigenvalue equation

H \left|\phi_n\right\rangle = E_n \left|\phi_n\right\rangle\,

is known as the time-independent Schrödinger equation. The time-dependent Schrödinger equation has solution

\left|\Psi(t)\right\rangle = C e^{-\frac{i}{\hbar}E_n t} \left|\phi_n\right\rangle\,,

where C\, is some constant. A general state \left|\Psi(t)\right\rangle\, can be decomposed into the eigenstates of H\,:

\left|\Psi(t)\right\rangle = \sum_n C_n(t) \left| \phi_n \right\rangle\,.

The time-dependent Schrödinger equation is then

i \hbar \frac{\partial}{\partial t} \left| \Psi(t) \right\rangle = i \hbar \frac{\partial}{\partial t} \sum_n C_n(t) \left| \phi_n \right\rangle = \sum_n C_n(t) H \left| \phi_n \right\rangle = \sum_n C_n(t) E_n \left| \phi_n \right\rangle\,.

Acting on this equation from the left by \left\langle \phi_m \right|\,, keeping in mind that \left\langle\phi_m|\phi_n\right\rangle = \delta_{mn}\, gives

i \hbar \frac{\partial}{\partial t} C_n(t) = E_n(t)\,,

which has solution

C_n(t) = C_{n}(0)e^{-\frac{i}{\hbar}E_n t}\,.

The full state is then

\left|\Psi(t)\right\rangle = \sum_n C_{n}(0)e^{-\frac{i}{\hbar}E_n t}\left|\phi_n\right\rangle\,.

Therefore, given knowledge of the initial conditions C_{n}(0)\, and the eigenvectors of H\,, we know the value of \left|\Psi(t)\right\rangle\, at all times.

Symmetries

Note that dynamics are unchanged under time reversal, i.e., t \to -t\, and \Psi(t) \to \Psi^*(t)\,. The heat equation does not have this symmetry.

back to Schrödinger picture
on to Poisson bracket

References

[1] [2]

  1. ↑ E. Schrödinger (1926). "An Undulatory Theory of the Mechanics of Atoms and Molecules". Phys. Rev. 28. DOI:10.1103/PhysRev.28.1049. 
  2. ↑ E. Schrödinger (1926). "Quantification of the eigen value problem". Annalen der Physik, 80 (13). 
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