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The TimeDependent Schrödinger Equation
We are now ready to consider the timedependent Schrödinger
equation. Although we were able to derive the singleparticle
timeindependent Schrödinger equation starting from the classical
wave equation and the de Broglie relation, the timedependent
Schrödinger equation cannot be derived using elementary methods and
is generally given as a postulate of quantum mechanics. It is
possible to show that the timedependent equation is at least reasonable if not derivable, but the arguments are rather involved
(cf. Merzbacher [2], Section 3.2; Levine
[3], Section 1.4).
The singleparticle threedimensional timedependent Schrödinger
equation is

(21) 
where V is assumed to be a real function and represents the
potential energy of the system (a complex function V will act as a
source or sink for probability, as shown in Merzbacher
[2], problem 4.1). Wave Mechanics is the
branch of quantum mechanics with equation (21) as its
dynamical law. Note that equation (21) does not yet
account for spin or relativistic effects.
Of course the timedependent equation can be used to derive the
timeindependent equation. If we write the wavefunction as a product
of spatial and temporal terms,
,
then equation (21) becomes

(22) 
or

(23) 
Since the lefthand side is a function of t only and the right hand
side is a function of
only, the two sides must equal a constant.
If we tentatively designate this constant E (since the righthand side
clearly must have the dimensions of energy), then we extract two
ordinary differential equations, namely

(24) 
and

(25) 
The latter equation is once again the timeindependent Schrödinger equation.
The former equation is easily solved to yield

(26) 
The Hamiltonian in equation (25) is a Hermitian operator,
and the eigenvalues of a Hermitian operator must be real, so E is
real. This means that the solutions f(t) are purely oscillatory,
since f(t) never changes in magnitude (recall Euler's formula
). Thus
if

(27) 
then the total wave function
differs from
only by a phase factor of constant magnitude.
There are some interesting consequences of this. First of all, the
quantity
is time independent, as we can easily
show:

(28) 
Secondly, the expectation value for any timeindependent operator is
also timeindependent, if
satisfies equation
(27). By the same reasoning applied above,

(29) 
For these reasons, wave functions of the form (27) are
called stationary states. The state
is
``stationary,'' but the particle it describes is not!
Of course equation (27) represents a particular solution
to equation (21). The general solution to equation
(21) will be a linear combination of these particular
solutions, i.e.

(30) 
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