3.3 the time-dependent schrödinger equation · the time-dependent schrödinger equation is linear...

3.3 The time-dependent Schrödinger equation

Slides: Video 3.3.1 Introduction to the time-dependent Schrödinger equation

Text reference: Quantum Mechanics for Scientists and Engineers

Chapter 3 introduction

The time-dependent Schrödinger equation

David MillerQuantum mechanics for scientists and engineers

3.3 The time-dependent Schrödinger equation

Slides: Video 3.3.2 Rationalizing the time-dependent Schrödinger equation

Text reference: Quantum Mechanics for Scientists and Engineers

Sections 3.1 – 3.2

The time-dependent Schrödinger equation

Rationalizing the time-dependent Schrödinger equation

Quantum mechanics for scientists and engineers David Miller

Relation between energy and frequency

The relation between energy

and frequency

for photons

is E h

Relation between energy and frequency

The relation between energy

and frequency

for photons

is E h

quantum mechanics

Rationalizing the time-dependent equation

We want a time-dependent wave equationfor a particle with mass m

with this relation between energy and frequency

We might also reasonably want it to have plane wave solutions

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

Rationalizing the time-dependent equation

Schrödinger postulated the time-dependent equation

Note that for a uniform potentiale.g., for simplicity

with and waves of the form

are indeed solutions

22 ,

, , ,2

tt V t t i

m t

rr r r

0V E 22 /k mE

exp exp exp expEt Eti t kz i kz i ikz

Rationalizing the time-dependent equation

In his time-dependent equation

Schrödinger chose a sign for the right hand sidewhich means that a wave with a spatial part

is definitely going in the positive z directionThat wave, including its time dependence would be of the form (for )

22 ,

, , ,2

tt V t t i

m t

rr r r

exp ikz

exp /i kz Et 0V

Compatibility with the time-independent equation

Before examining the time-dependent equation furtherfirst we should check that it is compatible with the time-independent equation

The time-independent equation could apply if we had states of definite energy E, an eigenenergy

Suppose we had some corresponding eigenfunctionso that

r

2

2

2V E

m r r r r

Compatibility with the time-independent equation

As it standsthis solution is not a solution of the time-dependent equation

Putting in here for does not workbecause has no time-dependence

the right hand side is zerowhereas it should be

how do we resolve this?

r

22 ,

, , ,2

tt V t t i

m t

rr r r

r ,t r r

E r

Compatibility with the time-independent equation

Suppose that, instead of proposing the solutionwe propose

r , exp /t iEt r r

2

2 , ,2

t V tm

r r r

, exp /t iEt r r so solves the time-independent Schrödinger equation

2

2 exp / exp /2

iEt V iEtm

r r r

2

2 exp /2

V iEtm

r r r exp /E iEt r

,E t r

Compatibility with the time-independent equation

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

substitutingin the time-dependent equation gives

r

, exp /t iEt r r

22 ,

, ,2

tt V t i

m t

rr r r

, exp /t iEt r r so solves the time-dependent Schrödinger equation

exp /i iEtt

r

exp /i iEtt

r exp /Ei i iEt r

,E t r

Compatibility with the time-independent equation

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

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

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

thenis a solution of both the time-independent and the time-dependent Schrödinger equations

making these two equations compatible

r

, exp /t iEt r r

exp /iEt

r

Oscillations and time-independence

If we propose a solution

to a time-independent problem can this represent something that is stable in time?

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

e.g., probability density

, exp /t iEt r r

2 2, exp / exp /t iEt iEt r r r r

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

3.3 The time-dependent Schrödinger equation

Slides: Video 3.3.6 Linear superposition

Text reference: Quantum Mechanics for Scientists and Engineers

Section 3.4 – 3.5

The time-dependent Schrödinger equation

Linear superposition

Quantum mechanics for scientists and engineers David Miller

Linearity of Schrödinger’s equation

The time-dependent Schrödinger equation is linear in the wavefunction

One reason is that no higher powers of appear anywhere in the equation

A second reason is that appears in every termthere is no additive constant term anywhere

22 ,

, , ,2

tt V t t i

m t

rr r r

Linearity of Schrödinger’s equation

Linearity requires two conditionsobeyed by Schrödinger’s time-dependent equation

1 - If is a solution, then so also is a , where a is any 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) constantsis also a solution

a b

, , ,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 solutionsis the property of

linear superpositionTo emphasize

linear superpositions of solutions of the time-dependent Schrödinger equation

are also solutions

, , ,c a a b bt c t c t r r r

Time-dependence and expansion in eigenstates

We know thatif the potential V is constant in time

each of the energy eigenstateswith eigenenergy En

is separately a solution of the time-dependent Schrödinger equation

provided we remember to multiply by the right complex exponential factor

n r

, exp /n n nt iE t r r

Time-dependence and expansion in eigenstates

Now we also know that the set of eigenfunctions of problems we will consider is a complete set

so the wavefunction at can be expanded in them

where the an are the expansion coefficientsBut we know that a function that starts out as

will evolve in time as so, by linear superposition, the solution at time t is

0t ,0 n n

n

a r r

n r , exp /n n nt iE t r r

, , exp /n n n n nn n

t a t a iE t r r r

Time-dependence and expansion in eigenstates

Hence, for the case where the potential V does not vary in time

is the solution of the time-dependent equationwith the initial condition

Hence, if we expand the wavefunction at time in the energy eigenstates

we have solved for the time evolution of the state just by adding up the above sum

, , exp /n n n n nn n

t a t a iE t r r r

,0 n nn

a r r r

0t