ФИАН / Nefedyev_Ktp

{px} = 1 = [ˆpxˆ] = −i.

a(t) = r

x + i√2mω ,

2

mω

p

a (t) = r

2

x − i√2mω .

mω

p

a

a

{a(t)a (t)} = i,

H =

1

ω(a (t)a(t) + a(t)a (t)).

2

a˙ (t) = Ha(t)} = −iωa(t)

a˙ (t) ={{Ha (t)} = iωa (t)

a(t) = a+e−iωt

a (t) = a−eiωt.

x(t)

a±

1

1

x(t) =

√

(a(t) + a (t)) =

√

(a+e−iωt + a−eiωt),

2mω

2mω

x(t)

a+ = a− ≡ a,

1

(ae−iωt + a eiωt).

x(t) =

√

2mω

a

a

aˆ† aˆ

[ˆaaˆ†] = 1.

En = ω

n + 2

,

1

Hˆ = 2ω aˆ†aˆ + aˆaˆ† = ω

aˆ†aˆ + 2

= ω

Nˆ + 2

,

1

1

1

ˆ

ˆ †

†

.

[Naˆ] = −a,ˆ

[Naˆ

] = aˆ

ˆ

ˆ

ˆ

− 1)a|ni

Naˆ|ni = aˆN|ni + [Naˆ]|ni = (n

ˆ †

† ˆ

ˆ

†

|ni,

Naˆ

|ni = aˆ N|ni + [Naˆ]|ni = (n + 1)a

aˆ†

aˆ

ˆ

X

†

†

|n − 1i|

2

2

,

n = hn|N|ni =

hn|aˆ

|kihk|a|ni = |hn|aˆ

= |hn − 1|aˆ|ni|

a†

0

i

|

1

(a†)n|0i, hn|ni = 1.

|ni =

√

n!

hx|0i ≡ ψ0(x)

|0i

r

x + √2mω

∂x

ψ0(x) = 0

2

mω

1

∂

√2

ξ +

∂ξ

ψ0(ξ) = 0,

ξ = √mωx.

1

∂

2

1

2

=

mω

1/4

ψ0(x) = N e−ξ

/2 = N e−

2 mωx

,

N

.

π

ψn(x)

n

√2

ξ − ∂ξ

n

ψn(ξ) = √n!

ψ0(ξ).

1

1

∂

f(ξ)

/2 ∂ξ e−ξ

/2f(ξ) .

ξ − ∂ξ f(ξ) = −eξ

∂

2

∂

2

1

2

dn

2

1

2

dn

2

ψn(ξ) =

√

(−1)neξ

/2

e−ξ

/2

ψ0(ξ) = N

√

(−1)neξ

/2

e−ξ

dξn

dξn

2nn!

2nn!

ψ (ξ) =

N

e−ξ2/2H

(ξ).

n

√2nn!

n

mω

1/4

mωx2

ψn(x) =

√

1

exp −

Hn

√mωx .

π

2

2nn!

α

α

1

2

∞

αn

|αi = e− 2 |α|

X

√

|ni,

n=0

n!

n

∞

hα|α′i = exp

−2

|α|2 + α α′ − 2

|α′|2

,

1

1

Iˆ = Z |αidπαhα| = Z |αi 2πi hα|,

2

dαdα

α

a

a†

x = √2mω (a† + a), p = ir

(a† − a).

2

1

mω

hα|a† = α hα|,

1

(α + α ),

hα|x2|αi

1

((α + α )2 + 1),

hα|x|αi =

√

=

2mω

2mω

hα|p|αi = ir

2 2

(α − α), hα|p2|αi = 2 (1 − (α − α)2).

mω

mω

x = hα|x2|αi − hα|x|αi2 = √2mω ,

p =

hα|p2|αi − hα|p|αi2 = r

2

p

1

p

mω

~/2

α

p · x

Pˆµ =

k

1 = Pσ=1,2 1/2

1

=

k

kµ

Cˆ†(k, 1)Cˆ(k, 1) + Cˆ†(k, 2)Cˆ

(k, 2) + 1

σ=1,2 kµ C†(k, σ)C(k, σ) +

2

X X

X

h

i

→

k

kµ hCˆ†(k, 1)Cˆ(k, 1) + Cˆ†(k, 2)Cˆ(k, 2) + Cˆ†(k, 3)Cˆ(k, 3) − Cˆ†(k, 0)Cˆ(k, 0) + 1i

X

3

3

= −

k σ=0 kµ hCˆ†(k, σ)Cˆ(k, σ) + 1i

→ −

k

σ=0 kµCˆ†

(k, σ)Cˆ(k, σ),

X X

X X

1

d3x A˙ 2

+ [ A]2

H = 2 Z

q = A(x),

˙

p = A(x) = −E(x).

˙

(3)

(x − y).

[pi(x, t)qj (y, t)] = [Ai(x, t)Aj(y, t)] = −iδij δ