Solutions of the Time-Dependent Schrodinger Eqn.
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Transcript of Solutions of the Time-Dependent Schrodinger Eqn.
3.3 The time-dependent Schrödinger equation
Slides: Video 3.3.4 Solutions of the time-dependent Schrödinger equation
Text reference: Quantum Mechanics for Scientists and Engineers
Section 3.3
The time-dependent Schrödinger equation
Solutions of the time-dependent Schrödinger equation
Quantum mechanics for scientists and engineers David Miller
Contrast to classical wave equation
The common classical wave equation has a different form
for which
would also be a solutionNote the classical equation has a second time derivative
as opposed to the first time derivative in Schrödinger’s time-dependent equation
2 22
2 2
k fft
expf i kz t
Schrödinger’s complex waves
Note that Schrödinger’s use of a complex wave equation
with the “i” on the right hand sidemeans that generally the wave is required to be
a complex entityFor example, for
though is a solutionis not a solution
22 ,
, , ,2
tt V t t i
m t
r
r r r
0V exp /i kz Et
sin /kz Et
Wave equation solutions
With the classical wave equationif at some time we see a particular shape of wave
e.g., on a string
Wave equation solutions
With the classical wave equationif at some time we see a particular shape of wave
e.g., on a string
we do not know if it is going to the right f z ct
Wave equation solutions
With the classical wave equationif at some time we see a particular shape of wave
e.g., on a string
we do not know if it is going to the rightor to the left
or even some combination of the two
f z ct g z ct
Time evolution from Schrödinger’s equation
In Schrödinger’s equation, for a known potential V
if we knew the wavefunction at every point in space at some time to
we could evaluate the left hand side of the equation at that time for all r
so we would know for all rso we could integrate the equation to deduce at all future times
22 ,
, , ,2
tt V t t i
m t
rr r r
, ot r
, /t t r
,t r
Time evolution from Schrödinger’s equation
Explicitlyknowing we can calculate
that is, we can know the new wavefunction in space at the next instant in time
and we can continue on to the next instantand so on
predicting all future evolution of the wavefunction
, /t t r
,
, ,o
o ot
t t t tt
r
r r