4.1 Time evolution of superpositions

Slides: Video 4.1.6 The coherent state

Text reference: Quantum Mechanics for Scientists and Engineers

Section 3.6 (“Coherent state”)

Time evolution of superpositions

The coherent state

Quantum mechanics for scientists and engineers David Miller

The coherent state

The coherent state for a harmonic oscillator of frequency is

where

and the are the harmonic oscillator eigenstates

0

1, exp2N Nn n

n

t c i n t

exp!

n

Nn

N Nc

n

n

The coherent state

Incidentally, note that for the expansion coefficients cNn

This is the Poisson distribution from statisticswith mean N and standard deviation

We will make no direct use of this herebut in the end it explains, e.g., the Poissonian distribution of photons in a laser beam

2 exp!

n

Nn

N Nc

n

N

Coherent state

Coherent state oscillationswith

0

,

1exp2

N

Nn nn

t

c i n t

exp!

n

Nn

N Nc

n

Coherent state

Coherent state oscillationswith

z

0

,

1exp2

N

Nn nn

t

c i n t

exp!

n

Nn

N Nc

n

1N 2,N t

Coherent state

Coherent state oscillationswith

z

0

,

1exp2

N

Nn nn

t

c i n t

exp!

n

Nn

N Nc

n

3N 2,N t

Coherent state

Coherent state oscillationswith

z

0

,

1exp2

N

Nn nn

t

c i n t

exp!

n

Nn

N Nc

n

10N 2,N t

Coherent state

Coherent state oscillationswith

z

0

,

1exp2

N

Nn nn

t

c i n t

exp!

n

Nn

N Nc

n

100N 2,N t

Finite well superposition

Make an equal superposition of the first three states of a finite potential wellas in our previous example

Because the energies are not rationally relatedthe superposition never

repeats

18oV E

00.663

2.603

5.609

Finite well superposition

Make an equal superposition of the first three states of a finite potential wellas in our previous example

Because the energies are not rationally relatedthe superposition never

repeatse.g., in the probability

density in time

18oV E

00.663

2.603

5.609