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We know time independent Schrodinger equation, where the wave function is stationary in time and varies in space. Simple examples are, particle in an infinite potential well, and the Hydrogen atom, which is a particle in a time independent, radially attractive electric field.

What I wonder and would like to know is that, is there any real world situation where the wave function is stationary in space and varying in time, something on the lines of "space-independent Schrodinger equation", if it exists. (I don't know its formula or its existence, even in theory).

Something like $$\Psi(x,t) = \psi(t)e^{-ikx}$$. I'd like to know if any simple examples exist and also its physical interpretations. I am just super curious. (The observable still being assumed as bounded self adjoint linear operators), everything about QM still intact.

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    $\begingroup$ a wave-function which is constant in space can only be the zero function because of the normalisation/ boundary conditions. $\endgroup$ – Phoenix87 Jan 3 '15 at 13:19
  • $\begingroup$ @Phoenix87 : I don't constant, I mean stationary, like appearing as a factor $e^{-ikx}$. I hope we can imagine or cook up some boundary conditions. I am not sure here, would like to know how I can be wrong in this quest. $\endgroup$ – Rajesh Dachiraju Jan 3 '15 at 13:23
  • $\begingroup$ @Phoenix87 : Please feel free to throw up particle formalism, if need be and if it helps. $\endgroup$ – Rajesh Dachiraju Jan 3 '15 at 13:25
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    $\begingroup$ You'll have solutions of that form for a free particle $\endgroup$ – Phoenix87 Jan 3 '15 at 13:47
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    $\begingroup$ The wavefunction you write is not square integrable, therefore not a suitable wavefunction for a quantum mechanical system. $\endgroup$ – yuggib Jan 3 '15 at 14:43
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Wait a minute.

You are saying:

We know time independent Schrodinger equation, where the wave function is stationary in time and varies in space. Simple examples are, particle in an infinite potential well, and the Hydrogen atom, which is a particle in a time independent, radially attractive electric field.

That's a bit wrong. In both cases, the energy eigenstates (the solutions to the Schrodinger equation) happen to be stationary states. This means:

1) Physically, that any observable associated with them (which is going to depend on $|\Psi|^2$) is time independent.

2) Mathematically, that the time dependence is enclosed in a phase factor, such that you can write the total wavefunction as $\Psi(\mathbf{x},t) = \psi(\mathbf{x})\,e^{-iEt/\hbar}$, where the $-$ sign in the exponent can also be a $+$ sign depending on convention.

This boils down to the fact that the Hamiltonian is time-independent: all systems with a time-independent hamiltonian (a counterexample could be an electron in a time-varying electric field, generated by a laser) will yield stationary states. It comes from the fact that you can solve the TDSE (Time Depedent) via separation of variables and obtaining a time-free differential equation called TISE (Time Independent) depending on the potential and a differential equation for time only that will integrate to give you a complex exponential (i.e. a pure phase factor).

This is to say that the wavefunction is still varying in time, but only in its phase factor: it makes all observables time-independent.

Now to answer your question: it's impossible.

If you had indeed $\Psi(x,t) = \psi(t)e^{-ikx}$, its norm $|\Psi|^2 = |\psi(t)|^2 $ would need to be constant to ensure that probability is conserved. This can only happens if $\psi(t)$ were a pure phase factor, $\propto e^{-i\omega t}$.

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