Redshift Academy


Position Creation and Annihilation Operators for Bosons
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Notation:  Capital Ψ is used to denote field operators and ψ is used
for wavefunctions.

Field operators create or destroy a particle at a particular point in space.  
Field operators act by applying the Fourier transform to the creation and 
annihilation operators.  

Ψ(x) = Σ_kα(k)exp(ikx) where α(k) is equivalent to the amplitude term in QM

and,

Ψ^*(x) = Σ_kα^*(k)exp(-ikx)

Compare these to the QM case:

ψ(x) = ΣA_mψ_m
       ^m

Where ψ_m are othornormal basis vectors of the form exp(ikx) and A_m are the 
Fourier coefficients.  Therefore, in QFT the Fourier coefficients are 
replaced by the creation and annihilation operators.

Now introduce time:

Ψ(x,t) = Σ_ka^-exp(i(kx - ωt))  ... annihilates a particle at position x. 

and,

Ψ^†(x,t) = Σ_ka⁺exp(-i(kx - ωt))  ... creates a particle at position x. 

Ψ(x,t) and Ψ^†(x,t) are quantum fields.

Summary:

a⁺(k) - creation operator for momenta, k
Ψ^†(x) - creation operator for position, x
a^-(k) - annihilation operator for momenta, k
Ψ(x) - annihilation operator for position, x

Examples:

Pre-existing photon, n(5) = 1
                   
                    |0 0 0 0 1 0 0> 

Create a new photon in momentum state k = 3 at x.  Thus, 
n(3) -> 1, n(5) = 1

Σ_ka⁺(3)exp(-i3x)|0 0 0 0 1 0 0> => e^-i3x|0 0 1 0 1 0 0>

Create another new photon in momentum state k = 5 at the same x. 
Thus, n(5) -> 2

Σ_ka⁺(5)exp(-i5x)|0 0 1 0 1 0 0> => √2e^-i5x|0 0 1 0 2 0 0>  

The factor of √2 comes from a⁺|n> = √(n+1)|n+1>

Creation and annihilation operators on bra vectors:

The rule for the creation and annihilation operators when they 
operate on bra vectors is the opposite of the case for ket 
vectors.  Thus, the creation operator acts on a bra as an 
annihilation operator and vice versa.

<n|a⁺ => <n-1|√n

and

<n|a^- => <n+1|√n+1

Consider the following scattering processes:

t        k₂
|         \ 
|          \ |
|           \| g = coupling contant that measures the strength of
|           /|       the scattering (measured experimentally).
|          / |
|         /  |
|       k₁
|
 -------------------- x
            x=0
 
Ψ^†Ψ|k₁> => |k₂> 

The probability that k₁ results in k₂ is given by:

|<k₂|Ψ^†Ψ|k₁>|²

We will just calculate the |...|:

<k₂|g∫dtΣ_da⁺(d)exp(iω₂t)Σ_ea^-(f)exp(-iω₁t)|k₁> after setting x = 0

<k₂|g∫dta⁺(k₂)exp(iω₂t)a^-(k₁)exp(-iω₁t)|k₁>

<0|g∫dtexp(i(ω₂ - ω₁)t)|0>

<0|gδ(ω₂ - ω₁)|0>

∴ hω₂ = hω₁ ... conservation of energy 


        k₂     k₃
         \    /
          \  /
           \/
           / g
          / 
         /  
       k_i

Follow the same procedure as above:

Ψ^†Ψ^†Ψ|k₁> => |k₂k₃>

<k₂k₃|Ψ^†Ψ^†Ψ|k₁>

<k₂k₃|g∫dxΣ_da⁺(d)exp(-ik₃x)Σ_ea⁺(e)exp(-ik₂x)Σ_fa^-(f)exp(ik₁x)|k₁> after setting t = 0

<k₂k₃|g∫dxa⁺(k₃)exp(-ik₃x)a⁺(k₂)exp(-ik₂x)a^-(k₁)exp(ik₁x)|k₁>

<0| = 1 when k₁ - k₂ - k₃ or 0 otherwise |0>

∴ hk₁ - hk₂ - hk₃ = 0 ... conservation of momentum

Time Dependent Schrodinger equation for a field:

Differentiate, Ψ w.r.t. t and x:

∂Ψ/∂t = Σ_k(-iω)a^-(k)exp(i(kx - ωt))

∂Ψ/∂x = Σ_k(ik)a^-(k)exp(i(kx - ωt))

∂²Ψ/∂x² = Σ_k(ik)²a^-(k)exp(i(kx - ωt))

So we can write:

∂Ψ/∂t/∂²Ψ/∂x² = Σ_k(-iω)a^-(k)exp(i(kx - ωt))/Σ_k(ik)²a^-(k)exp(i(kx - ωt))

            = (-iω)/(ik)²

Which leads to:

 (ik)²∂Ψ/∂t = (-iω)∂²Ψ/∂x²

    k²∂Ψ/∂t = (iω)∂²Ψ/∂x²

Now E = hω = p²/2m = h²k²/2m

Therefore k² = 2mω/h

(2mω/h)∂Ψ/∂t/∂ = (iω)∂²Ψ/∂x²

∂Ψ/∂t =(ih/2m)∂²Ψ/∂x²

Note: In both cases we could have equally integrated over 
position/time to demonstrate conservation of momentum and 
conservation of energy respectively.

What is the meaning of ∫Ψ^†(x)Ψ(x)dx ?

circle with length L

 = ∫_LΣ_ma⁺(m)e^-imxΣ_na^-(n)e^inxdx

 = ∫_LΣ_ma⁺(m)Σ_na^-(n)e^i(n-m)xdx
 
 = LΣ_ma⁺(m)a^-(m)  - occupation number is a⁺(m)a^-(m)

 = the number of particles

 Ψ^†(x)Ψ(x) is interpreted as the density of particles at x

Position Creation and Annihilation Operators for Fermions
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Bosons can occur in the same state state; Fermions cannot (Pauli
Exclusion Principle).  Fermions can only have 0 or 1 in a state. We
define new position creation and annihilation operators as:

Ψ(x,t) = Σ_kc^-exp(i(kx - ωt)) 

and,

Ψ^†(x,t) = Σ_kc⁺exp(-i(kx - ωt)) 

Heisenberg Uncertainty Principle in QFT
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Analagous to the relationship in QM, the position creation and 
annihilation operators are Fourier transforms of each other.  
Thus : 

Ψ(x)^± = Σ_ka(k)^±exp(±ikx)

and

a(k)^± = ΣΨ(x)^±exp(±ikx)

This shows that to create a state of definite position, it is necessary 
to sum over many momentum states. Conversely, to create a state 
of definite momentum it is necessary to sum over many position 
states.