rationalizing the time-dependent schrodinger eqn

8/10/2019 Rationalizing the Time-Dependent Schrodinger Eqn.

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3.3 The time-dependentSchrödinger equation

Slides: Video 3.3.2 Rationalizing the

time-dependent Schrödingerequation

Text reference: Quantum Mechanicsfor Scientists and Engineers

Sections 3.1 – 3.2



The time-dependent Schrödin

equation

Rationalizing the time-depenSchrödinger equation

Quantum mechanics for scientists and engineers Da



Relation between energy and frequen

The relation between

energy

and

frequency

forphotons

is E h



Relation between energy and frequen

The relation between

energy

andfrequency

forphotons

is E h

quantum mechanics



Rationalizing the time-dependent equatio

We want a time-dependent wave equation

for a particle with mass m

with this relation between enerfrequency

We might also reasonably want it to have plane wsolutions

e.g., of the formwhen we have some specific energy E

and when we are in a uniform potential

E h

exp i kz t




Schrödinger postulated the time-dependent equ

Note that for a uniform potential

e.g., for simplicitywith and

waves of the form

are indeed solutions

2

2 ,

, , ,2

t

t V t t im t

r

r r r

0V E

22 /k mE

exp exp exp ex Et Et

i t kz i kz i




In his time-dependent equation

Schrödinger chose a sign for the right hand sid

which means that a wave with a spatial part

is definitely going in the positive z directiThat wave, including its time dependenwould be of the form (for )

22

,

, , ,2

t t V t t i

m t

r

r r r

exp ikz

exp /i kz Et

0V



Compatibility with the time-independent e

Before examining the time-dependent equation f

first we should check that it is compatible with

time-independent equationThe time-independent equation could apply if we

states of definite energy E , an eigenenergySuppose we had some corresponding eigenfu

so that

2

2

2V E

m r r r r




As it stands

this solution is not a solution of the time

dependent equation

Putting in here for does not work

because has no time-dependencethe right hand side is zero

whereas it should be

how do we resolve this?

r

2

2 ,

, , ,2

t t V t t i

m t

r

r r r

r ,t r

r

E r




Suppose that, instead of proposing the solution

we propose , exp /t iEt r r

2

2 , ,2

t V t m

r r r

, exp /t iEt r r

so solves ttime-independent Schrödinger equat

2

2 exp / exp /2

iEt V iEt m

r r r

2

2 exp /2

V iEt m

r r r e E r

, E t r




Similarly, knowing that solves the time-independent equation with energy E

substitutingin the time-dependent equation gives

r

, exp /t iEt r r

2

2 ,, ,2

t t V t im t

rr r r

, exp /t iEt r r

so solves ttime-dependent Schrödinger equatio

ei t

r

exp /i iEt t

r exp /

E i i iEt

r




So every solution of the time-independentSchrödinger equation, with eigenenergy E

is also a solution of the time-dependent equaas long as we always multiply it by a factor

If is a solution of the time-independent Schequation, with eigenenergy E

thenis a solution of both the time-independen

and the time-dependent Schrödinger equmaking these two equations compatib

r

, exp /t iEt r r

r



Oscillations and time-independence

If we propose a solution

to a time-independent problemcan this represent something that is stable i

Yes! - measurable quantities associated with this are stable in time!

e.g., probability density

, exp /t iEt r r

2

, exp / exp /t iEt iEt r r r

rationalizing the time-dependent schrodinger eqn

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