linear superpositions

7/26/2019 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



The time-dependent Schrödin

equation

Linear superposition

Quantum mechanics for scientists and engineers Da



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



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



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



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




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




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

linear superpositions

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