2015-9-20 chang-kui duan, institute of modern physics, cupt 1 harmonic oscillator and coherent...

23/4/19 Chang-Kui Duan, Institute of Modern Physics, CUPT 1

Harmonic oscillator and coherent states

Reading materials:1. Chapter 7 of Shankar’s PQM.

1. Energy eigen states by algebra method2. Wavefunction3. Coherent state4. The most classical quantum system


Algebra method for eigen states

2 21 1

2 2H P kX

m

Defining dimensionless coordinate

1 42

1 42, so

1

Y mk X

Q mk P

position and momentum are now on equal foot.

Classical dynamics: y q

q y

The Hamiltonian of a harmonic oscillator is

2 2 ,2

H Y Q

where k m

Cyclic trajectory in the phase space.

y

q


Quantum mechanics: ,X P i Let us “rotate” the coordinates by an imaginary angle so that the cyclic rotation in the phase space is automatically taken into account by the transformation

† and 2 2

Y iQ Y iQa a

†, 1a a

† † †1 1

2 2H a a aa a a

or ,Y Q i

Note: Do you still remember how to write a circularly polarized light (whose electric field rotates) in terms of linear polarization?

The Hamiltonian becomes

How to get the eigen states and the eigen values?

† and can be viewed as "position" and "momentum" in thecoordinates of the phase space rotated by an imaginary angle.a a


2 2 02

H Y Q

So there must be a ground state 00 0H E

1. The energy spectrum must be lower bounded, for

2. There should be no continuum state, i.e., the eigenstate wavefunction should be normalized to one.

1 Otherwise the energy of the state cannot be finite due to the infinite potential diverging at the remote positions.

3. Theorem: In one-dimension space, a discrete state cannot be degenerate (see Shankar, PQM page 176 for a proof).

A few general properties to be used:


If there is an eigenstate H E E E

Ha E aH a E E a E

,H a a

† †,H a a

† † † †Ha E a H a E E a E

i.e, is also an eigen state with energy a E E

†i.e, is also an eigen state with energy a E E Repeating the process, we get a series of eigen states

† † †, , , , , , ,aa E aa E a E E a E a a E

Their energies form an equally space ladder, 2 , , , , 2 ,E E E E E


The energy ladder has to be lower bounded, so we must have

0 0a So the ground state energy is given by

†0

1 1 10 0 0 i.e.,

2 2 2H a a E

†

0

0 , where is a normalization factor

1 2

n

n n

n

n

n C a C

H n E n

E E n n

All the other eigen states can be obtained by


†

†

10

!

1

1 1

nn a

n

a n n n

a n n n

† † †Using , 1 and 0 0 :a a aa a a a

2 †0 0 1 1 !nn

n nn n C a a C n

matrix forms in the eigen state basis:

† †0 1 0aa a a † † 0 2! 0aaa a

† † † 0 3! 0aaaa a a

†

1 11 and 1

n na n n n a n n n

0 1 0 0

0 0 2 0

0 0 0 3

0 0 0 0

a


matrix forms in the eigen state basis:

0 1 0 0

1 0 2 01

0 2 0 32

0 0 3 0

Y

The dimensionless position and momentum operators are

† and 2 2

Y iQ Y iQa a

† †

and 2 2

a a a aY Q

i

0 1 0 0

1 0 2 01

0 2 0 32

0 0 3 0

i

i i

Q i i

i


WavefunctionsLet us first consider the ground state:

0 0a i.e., 0 02

Y iQ

In the dimensional coordinates: 0

10 0

2 2

Y iQ dy y y

dy

0 0

dy y y

dy

The solution is 20 1 4

1exp 2y y

So the ground state wavefunction in the real space is:

1 4

2 20 2

exp 2mk

x mk x

A nice property of Gaussian function is that its Fourier transformation is also a Gaussian function. The wavefunction in the q-representation would have the same form (remember Y and Q are inter-exchangeable.

A wavepacket centered at the potential minimum.


Now consider the excited states:

†10

!

nn a

n

†

2

Y iQa

Thus the wavefunction in real space is

21exp 2

2 !

n

nn

dy y y

dyn

21

exp 22 !

nn

H y yn

where is the th order Hermite polynomial. nH y n

The above equation actually defines the generation of Hermite polynomials.

The wavefunction in the momentum-representation is

21exp 2

2 !

n

nn

dq i iq q

dqn

2exp 2

2 !

n

nn

iH q q

n

defining a nice property of the F.T. of Hermite polynomials.


Regarded as bosonNow that all the eigen states of a harmonic oscillator are equally spaced., we can take the rising from one state to the next one as the addition of one particle with the same energy to a mode. The ground state contains no particle and hence is the vacuum state. Simple one mode can have an arbitrary number of particles, this particle is a boson.

0 Vacuum state

2

The state with n bosons, called a Fock staten

The operator annihilating one bosona

The operator creating one boson†a

The boson particle number operator†n a a

The energy of the boson

The energy of the vacuum


Coherent stateA coherent state of a harmonic oscillator is defined as

†exp 0 where is a complex number.C Ca C a C

In terms of the position and momentum operators, it is

2 0 where and .i ri C Y C Qr iC e C C C C


Coherent state in Fock state basis

To derive the wave function of the coherent state in the Fock state basis, we use the Baker-Hausdorff theorem

† *exp 0C Ca C a 0

nn

C n

† *, is a -number,Ca C a CC c

so the condition for the theorem is satisfied.

†2 0CC Ca C aC e e e

1

,2 if , , , , 0A BA B A Be e e e A A B A B B


Coherent state in Fock state basis *

2*

* 212!

C aC

e C a a

but 0 0a *

so 0 0C ae

†exp 2 exp exp 0C CC Ca C a

†exp 2 exp 0CC Ca

†2

† †2

0

0 1 02! !

nCa

n

C Ce Ca a n

n

* 2

0 !

nCC

n

CC e n

n

The expansion is


The state is normalized (of course) as can be checked directly:

**

, 0 ! !

m nCC

n m

C CC C e m n

m n

**

0

1!

n nCC

n

C Ce

n

Coherent state is an eigen state of the annihilation operator: Removing one boson does NOT change a coherent state!

* 2

0 !

nCC

n

Ca C e a n

n

†a C CHowever, adding one boson changes the state

* 2

0

1!

nCC

n

Ce n n

n

C C

Like being an eigen state of the position operator , a coherentstate can be viewed as an eigen state of the "position" operator inthe phase-space coordinates rotated by an imaginery angle.

x Xa


The expectation value of the boson number is

2†n C a a C C

Boson number distribution (Poisson distribution)

2

2

! !

n n

C nC np n n C C n e e

n n

p n

n


To understand the nature of the coherent state, let us consider first a few special case:

21. 0 , i.e., is pure imagineryii C YC e C

The real space wave function of this state is

20

ii C yC y e y

Or in the momentum representation

0 22

iqy

C C i

eq y dy q C

The state is the ground state with the momentum distribution shifted by

2 iC

y

q

2 iC


22. 0 , i.e., is realri C QC e C

The real space wave function of this state is

20

ii C qC q e q

The wave function in the momentum representation is

0 22

iqy

C C r

ey q dq y C

The state is the ground state with the real-space distribution shifted by

2 rC y

q

2 rC


3. For an arbitary complex number C

For more, do the homework.

2 0i ri C Y C QC e

Baker-Hausdorff theorem

1

,2 if , , , , 0A BA B A Be e e e A A B A B B

22 2 2i r i ri C Y C Q i C i C Y i C Qe e e e

The state is shifted from the groundstate in the phase space , by

2 ,r i

y q

C C y

q

2C


Quantum fluctuation

,Y Q i

According to Heisenberg principle

1 2Y Q

Fock states

0Q n Q n n Y n n Y n

† † † †2 2

† † † †2 2

1

2 21

2 2

aa a a aa a aY n Y n n n n

aa a a aa a aQ n Q n n n n

1 2 1 2Y Q n

† †

and 2 2

a a a aY Q

i


In particular, for the vacuum state

The vacuum state has minimum quantum fluctuation (the most classical state)

0 0 0 0 0 0 0Q Q Y Y † † † †

2 2

† † † †2 2

10 0 0 0

2 21

0 0 0 02 2

aa a a aa a aY Y

aa a a aa a aQ Q

1 2Y Q


Quantum fluctuation: Coherent stateFor a coherent state

†

22

a aQ C Q C C C C

i

†

22

a aY C Y C C C C

The center of the wavepacket in the phase space is at 2C


1 2Y Q

A coherent state has minimum quantum fluctuation.

† † † † 2

2 12

2 2

aa a a aa a aQ C C C

The variance can be obtained by calculating

† † † † 2

2 12

2 2

aa a a aa a aY C C C

This justify our viewing a coherent state as a wavepacket centered at a point in the phase space and a most classical state.


Completeness condition:

211C C d C

The coherent states for all complex numbers form a complete basis

2*

2 2

, 0

1 1

! !

m nC

n m

C CC C d C m n e d C

n m

2 2

2

* 2

2,2

C i m nm n n m

nn m

e C C d C e e d d

e d

,! n mn

Coherent states as a basis


2 2expC C C C

The Hilbert space has been expanded by a discrete set of states (the Fock states). But the complex numbers form a continuum. So the coherent states must be over complete, i.e., more than enough, since they are not orthogonal:


The most classical quantum system


We have seen that a coherent state can be viewed as a shift in the phase space from the vacuum state.

The vacuum state, of course, is also a coherent state.

What if we do the shift from a coherent state other than the vacuum?

† *, exp C ?C C C a C a

† †exp exp ?U C a C a Ca C a

Repeatedly using Baker-Hausdorff theorem:2 2 † †2 2C C C a C a Ca C aU e e e e e

2 2 † †2 2C C CC C a Ca C a C ae e e e e e

2 †22 2 C C C C a C C aCC C Ce e e e

* *, exp 2 2C C C C C C C C

It is still a coherent state, up to a trivial phase factor.


† * * *exp C exp 2 2C a C a C C C C C C

A shift in the phase space from a coherent state is still a coherent state, the total shift from the vacuum is just the sum of the two shifts.

Thus we can define a shift operator

† *expCD Ca C a

0CC D


Time evolution of a coherent stateIf we have an initial coherent state:

* 2

0 !

nCC

n

CC e n

n

The time evolution is simply

* 2 2

0 !

nCC in t i t

n

CC t e e n

n

2i t i tC t e Ce

It is a coherent state with its shift from the vacuum rotating in the phase space like a classical oscillator.

y

q

2C t

2C


Harmonic oscillator driven by a forceLet us consider the motion of a harmonic oscillator starting from the ground state

2 21 1

2 2H p kx exE t

m

† †1 2H a a a a t

In the form of boson operators:

Suppose the state at a certain time is tThe Schroedinger equation is

ti t H t t Consider an infinitesimal time increase

expt dt t iHdt t iHdt t


† †exp 1 1 1 2iHdt iHdt i a a dt i a a t dt

† †1 1 2 1i a a dt i a a t dt

† †exp 1 2 expi a a dt i a a t dt

The first term is a free evolution of the state.

The second term is the shift operator which shift a state in the phase space along the position axis.

†0 exp 1 2U t i a a dt

†1 expU t i a a t dt

The evolution from the initial state in a finite time is

0 1 0 1 0 1 0 10 0 0t U t U t U t dt U t dt U dt U dt U U


So, if the initial is a coherent state, say, the vacuum state, the state after a finite time of evolution is still a coherent state

0C C t

And we have the shift for an infinitesimal time increase to be

i dtdC t C t dt C t e C t i dt C t

i dtC t i dt

The equation of motion is: dC t i C t i

dt

i.e., r i i

i i r

C t C t

C t C t

The same as the classical eqns. for position and momentum!


Conclusion: A harmonic oscillator driven by a classical force from the ground state is always in a coherent state.

We have seen that the coherent state follows basically the equations for the classical eqns for position and momentum. It could be taken as a reproduction of the classical dynamics from quantum mechanics. The coherent state could be understood as classical particle, though it is quite a wavepacket (it is just so small that we had not enough resolution to tell it from a particle).

So, if we have only classical forces and harmonic oscillators, there is no way to obtain a “real” quantum state from the vacuum or the ground state. That is why we call harmonic oscillators the most classical quantum systems.

2015-9-20 chang-kui duan, institute of modern physics, cupt 1 harmonic oscillator and coherent...

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