The goal is to construct a set of quantum state for a harmonic oscillator of mass m and frequency ω resembling as much as possible its classical coun- terpart. The two basic requirements are

Luigi Pilo

1 Coherent States

The goal is to construct a set of quantum state for a harmonic oscillator of mass m and frequency ω resembling as much as possible its classical coun- terpart. The two basic requirements are

• the states must have minimal uncertainty, namely ∆x ∆p = ~/2;

• the time dependent average of coordinate and momentum operators must follow the classical evolution.

The fact that in the Heisenberg picture ˆ p(t) and ˆ q(t) have the same time evolution of the classical counterpart is not of much help for our goal; indeed, using parity (see appendix) for any eigenstate |n > of the harmonic oscillator we have that

< n|ˆ p(t)|n >=< n|ˆ q(t)|n >= 0 . (1) Classically, the motion is started displacing the oscillator from its equilibrium position with a non vanishing initial velocity. Let us try to mimic the same situation quantum mechanically; the equilibrium position is associated with the vacuum state |0 >, the displacement from the equilibrium postion is associated to a translation in space of a quantity c, the initial velocity with a translation in momentum space of a quantity b. We are then led to consider the following initial state |z >

|z >= e

e

|0 >= ˆ U (c, b)|0 > . (2) By construction, being ˆ U a unitary operator, < z|z >= 1.

Using the following commutation rule

[ˆ p, f (ˆ q)] = −i f

(ˆ q) , [ˆ q, f (ˆ p)] = i f

(ˆ p) , (3) we have that

e

q e ˆ

= ˆ q + e

[ˆ q , e

] = ˆ q + c ; (4)

1

One can check them by expanding f in a Taylor series and using, as many times as

required, [A, B C] = [A, B] C + B [A, C].

and

e

p e ˆ

= ˆ p + e

[ˆ p , e

] = ˆ p + b . (5) To simplify the notation it is convenient to introduce the dimensionless op- erators

P = ˆ p ˆ

(2m~ω)

, Q = ˆ  mω 2~



ˆ

q . (6)

Thus [ ˆ Q, ˆ P ] = i/2 and setting

α =  2mω

~



c , β =

 2

m~ω



; (7)

then

|z >= e

e

|0 >= ˆ U (α, β)|0 > ; e

P e ˆ

= ˆ P + β

2 , e

Q e ˆ

= ˆ Q + α 2 .

(8)

We can now show that |z > is an eigenstate of the annihilation operator ˆ

a = ˆ Q + i ˆ P ˆ

a|z >= ( ˆ Q + i ˆ P )e

e

|0 >= e

e

( ˆ Q + i ˆ P )e

e

|0 >

= e

( ˆ Q + i ˆ P + i β) e

|0 >= e

e

e

( ˆ Q + i ˆ P + i β

2 )e

|0 >

= e

e

( ˆ Q + i ˆ P + i β 2 + α

2 )|0 >= ˆ U (α, β)(ˆ a + i β 2 + α

2 )|0 >

= ˆ U (α, β)ˆ a|0 > + 1

2 (α + iβ)|z >= 1

2 (α + iβ)|z > .

(9) The state |z > is an eigenstate of ˆ a with a complex

eigenvalue z =

(α+iβ).

Using the properties of ˆ U is easy to compute the averages of momentum and coordinate operators

P =< z| ˆ ¯ P |z >=< 0|U

(α, β) ˆ P U (α, β)|0 >=< 0| ˆ P + β

2 |0 >= β

2 = Im(z) ; (10)

2

Note that ˆ a is not hermitian.

and in the same way

Q =< z| ˆ ¯ Q|z >= α

2 = Re(z) . (11)

Then the indetermination for momentum and coordinate are

∆Q

=< z|  ˆ Q − α 2



|z >=< 0|U

(α, β)  ˆ Q − α 2



U (α, β)|0 >

=< 0|  ˆ Q − α 2 + α

2



|0 >=< 0| ˆ Q

|0 >= mω

2~ < 0|ˆ q

|0 >= 1 4 .

(12)

The same result holds for momentum: ∆P

= 1/4. As a result

∆p

∆q

= 2~

mω 2m~ω ∆p

∆q

= ~

4 ; (13)

thus the state |z > has minimum uncertainty.

It very useful to find the decomposition of |z > in terms of the eigenstates {|n >} of the harmonic oscillator

|z >= X

n

< n|z > |n >= X

n

c

|n > ; (14) and

c

=< n|z >= 1

(n!)

< 0|ˆ a

|z >= z

(n!)

< 0|z > . (15) Thus

c

= z

(n!)

c

. (16)

To determine c

we impose that < z|z >= 1, 1 = X

n

|c

|

= X

n

|c

|

zz

n! = |c

|

e

⇒ c

= e

. (17) Setting the arbitrary phase to one, we get

|z >= e

X

n

z

(n!)

|n > . (18)

Using (18), time evolution is straightforward

|z, t > = e

|z >= X

n

c

e

|n >= X

n

c

e

|n >

= e

e

X

n

(e

z)

(n!)

|n >= e

|e

z > ;

(19)

then, modulo a phase factor, time evolution simply sends z into e

z. This result is very important and allows to compute easily time dependent aver- ages. Indeed, using (10-11) we get

Q(t) ¯ =< z, t| ˆ Q|z, t >=< |e

z| ˆ Q|e

z >= Re(e

z)

= 1

2 (α cos ωt + β sin ωt) (20)

P (t) ¯ =< z, t| ˆ P |z, t >=< |e

z| ˆ P |e

z >= Im(e

z)

= 1

2 (β cos ωt − α sin ωt) . (21)

The immediate consequence of (20-21) is that time averages behaves as the time evolution of the classical harmonic oscillator. In addition, because the

∆Q and ∆P do not depend on z, see for instance (12), and the time evolution basically changes z by a phase, we have that

∆Q

(t) = ∆Q

, ∆P

(t) = ∆P

; (22) thus

∆p

(t)∆q

(t) = ~

4 ; (23)

Finally it is interesting to compute the average and fluctuations of the number operator ˆ N = ˆ a

a

N =< z|ˆ ¯ a

a| >= |z|

; (24) and

∆N

= ˆ N

− N

=< z|ˆ a

aˆ a

a|0 > −|z|

= |z|

< z|aˆ a

|z > −|z|

= |z|

(1 + |z|

) − |z|

= −|z|

. (25)

The mean and the variance are the same of a Poisson distribution. Interpret- ing |c

|

as the probability P

distribution for n, the number of excitations, we get

P

= e

λ

n! , λ = |z|

, (26)

that is precisely a Poisson distribution of mean λ.

A Parity Selection Pules

Let Π be the parity operator and let be |S > any state with definite parity Π|S >= ±|S >=  |S > , 

= 1 . (27) Consider now an odd operator A

ΠAΠ = −A . (28)

An odd operator has zero average on a state with definite parity. Indeed, using Π

= 1

A =< S|A|S >=< S|Π ¯

A Π

|S >= 

< S|ΠAΠ|S >

=< S|ΠAΠ|S >= − < S|A|S >

⇒ ¯ A = − ¯ A or ¯ A = 0 .

(29)