Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Physics LibreTexts

3.10: 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, because 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 previous expression into Schrödinger’s equation ([e3.1]) 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 previous results to the case where H has continuous eigenvalues is straightforward.

If a dynamical variable is represented by some Hermitian operator A that commutes with H (so that it has simultaneous eigenstates with H), and contains no specific time dependence, then it is evident from Equations ([e4.157]) and ([e4.158]) 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 and Attributions

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


This page titled 3.10: Stationary States is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

Support Center

How can we help?