23.5 The Bargmann transform

The Stone von-Neumann theorem implies the existence of:

Definition

Bargmann transform

There is a unitary map called the Bargmann transform

B : H_{S} \to F

intertwining the Schr¨odinger representation and the Bargmann-Fock representation, $i . e .$ , with operators satisfying the relation

Γ_{S}^{'} (X) = B^{- 1} Γ_{B F}^{'} (X) B

for $X \in h_{3}$

In practice, knowing B explicitly is often not needed, since the representation independent relation

a = \frac{1}{2} (Q + i P)

can be used to express operators either purely in terms of a and $a^{†}$ , which have a simple expression

a = \frac{d}{d z}, a^{†} = z

in the Bargmann-Fock representation, or purely in terms of $Q$ and $P$ which have a simple expression

Q = q, P = - i \frac{d}{d q}

in the Schr¨odinger representation.

To compute the Bargmann transform one uses equation 23.11, for nonnormalizable continuous basis states $∣ δ_{u}$ , to get

⟨ q ∣ δ_{u} ⟩ = (\frac{ω}{π})^{\frac{1}{4}} e^{- \frac{u ^{2}}{2}} e^{- \frac{ω}{2} q^{2}} e^{2 ω \overline{u} q}

and

⟨ δ_{u} ∣ q ⟩ = (\frac{ω}{π})^{\frac{1}{4}} e^{- \frac{u ^{2}}{2}} e^{- \frac{ω}{2} q^{2}} e^{2 ω u q} (23.13)

The Bargmann transform is then given by

(B ψ) (z) = \int_{- \infty}^{+ \infty} ⟨ δ_{z} ∣ q ⟩ ⟨ q ∣ ψ ⟩ d q = (\frac{ω}{π})^{\frac{1}{4}} e^{- \frac{z ^{2}}{2}} \int_{- \infty}^{+ \infty} e^{- \frac{ω}{2} q^{2}} e^{2 ω z q} ψ (q) d q (23.14)

(here $ψ (q)$ is the position space wavefunction) while the inverse Bargmann transform is given by

(B^{- 1} ϕ) (q) = \frac{1}{π} \int_{C} ⟨ q ∣ δ_{u} ⟩ ⟨ δ_{u} ∣ ϕ ⟩ e^{- ∣ u ∣^{2}} d^{2} u = \frac{1}{π} (\frac{ω}{π})^{\frac{1}{4}} e^{- \frac{ω}{2} q^{2}} \int_{C} e^{- \frac{u ^{2}}{2}} e^{2 ω \overline{u} q} ϕ (u) e^{- ∣ u ∣^{2}} d^{2} u

(here $ϕ (z)$ is the Bargmann-Fock wavefunction).

As a check of equation 23.14, consider the case of the lowest energy state in the Schr¨odinger representation, where |0⟩ has coordinate space representation

ψ (q) = (\frac{ω}{π})^{\frac{1}{4}} e^{- \frac{ω q ^{2}}{2}}

and

(B ψ) (z) = (\frac{ω}{π})^{\frac{1}{4}} (\frac{ω}{π})^{\frac{1}{4}} e^{- \frac{z ^{2}}{2}} \int_{- \infty}^{+ \infty} e^{- \frac{ω}{2} q^{2}} e^{2 ω z q} e^{- \frac{ω q ^{2}}{2}} d q = (\frac{ω}{π})^{\frac{1}{2}} \int_{- \infty}^{+ \infty} e^{- \frac{z ^{2}}{2}} e^{- ω q^{2}} e^{2 ω z q} d q = (\frac{ω}{π})^{\frac{1}{2}} \int_{- \infty}^{+ \infty} e^{- ω (q - \frac{z}{2 ω})^{2}} d q = 1

which is the expression for the state |0⟩ in the Bargmann-Fock representation.

For an alternate way to compute the harmonic oscillator propagator, the kernel corresponding to applying the Bargmann transform, then the time evolution operator, then the inverse Bargmann transform can be calculated. This will give

⟨ q_{T} ∣ e^{- i H T} ∣ q_{0} ⟩ = \frac{1}{π} \int_{C} ⟨ q_{T} ∣ δ_{u} ⟩ ⟨ δ_{u} ∣ e^{- i T H} ∣ q_{0} ⟩ e^{- ∣ u ∣^{2}} d^{2} u = \frac{1}{π} e^{- i \frac{ω _{T}}{2}} \int_{C} ⟨ q_{T} ∣ δ_{u} ⟩ ⟨ δ_{u e^{- i ω_{T}}} ∣ q_{0} ⟩ e^{- ∣ u ∣^{2}} d^{2} u

from which 23.12 can be derived by a (dificult) manipulation of Gaussian integrals.

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23.5 The Bargmann transform

23.5 The Bargmann transform

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