linear superpositions
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3.3 The time-dependentSchrödinger equation
Slides: Video 3.3.6 Linear
superpositionText reference: Quantum Mechanics
for Scientists and EngineersSection 3.4 – 3.5
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The time-dependent Schrödin
equation
Linear superposition
Quantum mechanics for scientists and engineers Da
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Linearity of Schrödinger’s equation
The time-dependent Schrödinger equation is linethe wavefunction
One reason is that no higher powers of appanywhere in the equation
A second reason is that
appears in every there is no additive constant term anywh
22 ,
, , ,2
t t V t t i
m t
r
r r r
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Linearity of Schrödinger’s equation
Linearity requires two conditions
obeyed by Schrödinger’s time-dependent equ
1 - If is a solution, then so also is a , whany constant
2 - If a and b are solutions, then so also isA consequence of these two conditions is that
where ca
and cb
are (complex) constants
is also a solution
, , ,c a a b bt c t c t
r r r
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Linear superposition
The fact that
is a solution if a
and b
are solutions
is the property oflinear superposition
To emphasize
linear superpositions of solutions of thetime-dependent Schrödinger equation
are also solutions
, , ,c a a b b
t c t c t r r r
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Time-dependence and expansion in eigen
We know that
if the potential V is constant in time
each of the energy eigenstateswith eigenenergy E
n
is separately a solution of the time-depSchrödinger equation
provided we remember to multiply bright complex exponential factor
n r
, exp /n n n
t iE t
r r
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Time-dependence and expansion in eigen
Now we also know that the set of eigenfunctionsproblems we will consider is a complete set
so the wavefunction at can be expanded
where the an
are the expansion coefficients
But we know that a function that starts out as
will evolve in time asso, by linear superposition, the solution at t
0t
, 0n n
n
a r r
n
, exp /n n nt iE t r
, , exp /n n n n n
n n
t a t a iE t r r
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Time-dependence and expansion in eigen
Hence, for the case where the potential V does nin time
is the solution of the time-dependent equatiowith the initial condition
Hence, if we expand the wavefunction at time the energy eigenstates
we have solved for the time evolution of the stby adding up the above sum
, , exp /n n n n n
n n
t a t a iE t r r
,0n
r r
t
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