Physical Chemistry , 1st ed.

10.14 The Time-Dependent Schrödinger Equation

Although the time-independent Schrödinger equation is heavily utilized in this

chapter, it is not the fundamental form of the Schrödinger equation. Only sta-

tionary states—wavefunctions whose probability distributions do not vary over

time—provide meaningful eigenvalues using the time-independent Schrödinger

equation. There is a form of the Schrödinger equation that does include time.

It is called the time-dependent Schrödinger equation,and has the form

Hˆ(x,t) i





(x

t

,t)

(10.29)

where the x- and t-dependence on are written explicitly to indicate that 

does vary with time as well as position. Schrödinger postulated that all wave-

functions must satisfy this differential equation, and it is the last postulate we

will consider, if only briefly. This postulate is what establishes the prime im-

portance of the Hamiltonian operator in quantum mechanics.

One common way to approach equation 10.29 is to assume the separability

of time and position, similar to our separation ofx,y, and zin the 3-D box.

That is,

(x,t) f(t) 

(x) (10.30)

where part of the complete wavefunction depends only on time and part de-

pends only on position. Although it is fairly straightforward to derive, we will

omit the derivation and simply present the following statement of acceptable

solutions of(x,t):

(x,t) eiEt/

(x) (10.31)

where Eis the total energy of the system. This solution for the time-dependence

of a wavefunction places no restriction on the form of the position-dependent

function (x). With respect to wavefunctions, we are right back where

we started at the beginning of the chapter. With this assumption, the time-

dependence of the total wavefunction is rather simple in form, and the posi-

tion dependence of the total wavefunction needs to be considered for the sys-

tem of interest. Iftcan be separated from position in (x,t) and the wave-

function has the form from equation 10.31, then the time-dependent

Schrödinger equation simplifies into the time-independent Schrödinger equa-

tion, as shown below.

Example 10.17

Show that solutions for given in equation 10.31, when used in the time-

dependent Schrödinger equation, yield the time-independent Schrödinger

equation.

Solution

Using the separated solution for (x,t):

Hˆ[eiEt/

(x)] i





t

[eiEt/

(x)]

Taking the derivative of the exponential with respect to time [the derivative

does not affect (x), since it doesn’t depend on time]:

Hˆ[eiEt/
(x)] i
(x)





iE

[eiEt/]

308 CHAPTER 10 Introduction to Quantum Mechanics