- #26

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**interpretation of the quantized Schrödinger field**

Begin with

i∂

_{t}Ψ(x,t) = H

_{x}Ψ(x,t) ... [1] ,

and assume (for ease of notation) that H

_{x}has a discrete, nondegenerate spectrum. Let φ

_{E}(x) denote the corresponding eigenfunctions, so that

H

_{x}φ

_{E}(x) = Eφ

_{E}(x) ... [2] .

Since the eigenfunctions φ

_{E}(x) form a complete set, the field operator Ψ(x,t) can be expanded in terms of them. That is, we can write

Ψ(x,t) = ∑

_{E}a

_{E}(t)φ

_{E}(x) ... [3] ,

which when substituted into [1] in connection with [2] yields

ida

_{E}(t)/dt = Ea

_{E}(t) ,

so that

a

_{E}(t) = e

^{-iEt}a

_{E}... [4] .

Substituting this result back into [3] then gives

Ψ(x,t) = ∑

_{E}a

_{E}φ

_{E}(x)e

^{-iEt}... [5a] ,

and therefore,

Ψ

^{†}(x,t) = ∑

_{E}a

_{E}

^{†}φ

_{E}*(x)e

^{iEt}... [5b] .

___________________

Consider the two cases (as explained in the previous post):

(a) [Ψ(x,t),Ψ

^{†}(x',t)] = δ(x - x') , [Ψ(x,t),Ψ(x',t)] = [Ψ

^{†}(x,t),Ψ

^{†}(x',t)] = 0 ;

(b) {Ψ(x,t),Ψ

^{†}(x',t)} = δ(x - x') , {Ψ(x,t),Ψ(x',t)} = {Ψ

^{†}(x,t),Ψ

^{†}(x',t)} = 0 .

In light of the expressions [5a] and [5b] for Ψ(x,t) and Ψ

^{†}(x,t), the above two cases are seen to be equivalent to:

(a') [a

_{E},a

_{E'}

^{†}] = δ

_{EE'}, [a

_{E},a

_{E'}] = [a

_{E}

^{†},a

_{E'}

^{†}] = 0 ;

(b') {a

_{E},a

_{E'}

^{†}} = δ

_{EE'}, {a

_{E},a

_{E'}} = {a

_{E}

^{†},a

_{E'}

^{†}} = 0 .

Thus, the operators a

_{E}

^{†}and a

_{E}can be interpreted as

*creation*and

*annihilation*operators with respect to the eigenfunction "modes" φ

_{E}(x) of H

_{x}, where case (a) corresponds to bosonic excitations, and case (b) corresponds to fermionic excitations.

In particular we can write, for all E in the spectrum of H

_{x},

a

_{E}|0> = 0 ... [6a] ,

and

a

_{E}

^{†}|0> = |φ

_{E}> ... [6b] .

___________________

Consider the object Ψ

^{†}(x,0)|0>. From equations [5b] and [6b], we have

Ψ

^{†}(x,0)|0> = ∑

_{E}|φ

_{E}>φ

_{E}*(x) ,

and upon writing φ

_{E}*(x) = <φ

_{E}|x>, the last relation becomes

Ψ

^{†}(x,0)|0> = ( ∑

_{E}|φ

_{E}><φ

_{E}| ) |x> ;

that is,

Ψ

^{†}(x,0)|0> = |x> .

But from [5a] and [6a], we have

Ψ(x,0)|0> = 0 .

From these last two relations, in conjunction with the equal-time commutation or anticommutation relations for the field (i.e. cases (a) or (b) above), it follows that:

Ψ

^{†}(x,0) and Ψ(x,0) can be interpreted as

*creation*and

*annihilation*operators for a particle at the position x.