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Physics LibreTexts

4.12: Stationary States

( \newcommand{\kernel}{\mathrm{null}\,}\)

An eigenstate of the energy operator Hi/t corresponding to the eigenvalue Ei satisfies 

iψE(x,t,Ei)t=EiψE(x,t,Ei)

 

It is evident that this equation can be solved by writing
ψE(x,t,Ei)=ψi(x)eiEit/

where ψi(x) is a properly normalized stationary (i.e., non-time-varying) wavefunction. The wavefunction ψE(x,t,Ei) corresponds to a so-called stationary state, since the probability density |ψE|2 is non-time-varying. Note that a stationary state is associated with a unique value for the energy. Substitution of the above expression into Schrödinger's equation (137) yields the equation satisfied by the stationary wavefunction:

22md2ψidx2=[V(x)Ei]ψi

 

This is known as the time-independent Schrödinger equation. More generally, this equation takes the form
Hψi=Eiψi

where $H$ is assumed not to be an explicit function of $t$. Of course, the ψi satisfy the usual orthonormality condition:

ψiψjdx=δij

 

Moreover, we can express a general wavefunction as a linear combination of energy eigenstates:
ψ(x,t)=iciψi(x)eiEit/

where

ci=ψi(x)ψ(x,0)dx

Here, |ci|2 is the probability that a measurement of the energy will yield the eigenvalue Ei. Furthermore, immediately after such a measurement, the system is left in the corresponding energy eigenstate. The generalization of the above results to the case where $H$ has continuous eigenvalues is straightforward.

If a dynamical variable is represented by some Hermitian operator $A$ which commutes with $H$ (so that it has simultaneous eigenstates with $H$), and contains no specific time dependence, then it is evident from Eqs. (297) and (298) that the expectation value and variance of $A$ are time independent. In this sense, the dynamical variable in question is a constant of the motion.

Contributors

  • Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)


This page titled 4.12: Stationary States is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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