Defining the momentum space annihilation operator ^3 ϕ(𝐩)=∫⟨𝐩 |𝐫⟩ψ(𝐫) d^3 r derive the commutat

Step 1: **Define the momentum space annihilation operator** u000AThe momentum space annihilation op

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Defining the momentum space annihilation operator ${ }^3$ $$ \boldsymbol{\phi}(\mathbf{p})=\int\langle\mathbf{p} \mid \mathbf{r}\rangle \boldsymbol{\psi}(\mathbf{r}) d^3 r $$ derive the commutation (or anticommutation) relations for $\boldsymbol{\phi}(\mathbf{p})$ and $\boldsymbol{\phi}^{\dagger}(\mathbf{p})$. For the Bose-Einstein case, show that the mixed commutator of field operators in coordinate and momentum space is $$ \left[\boldsymbol{\phi}(\mathbf{p}), \boldsymbol{\psi}^{\dagger}(\mathbf{r})\right]=\langle\mathbf{p} \mid \mathbf{r}\rangle $$

   Defining the momentum space annihilation operator ${ }^3$
$$
\boldsymbol{\phi}(\mathbf{p})=\int\langle\mathbf{p} \mid \mathbf{r}\rangle \boldsymbol{\psi}(\mathbf{r}) d^3 r
$$
derive the commutation (or anticommutation) relations for $\boldsymbol{\phi}(\mathbf{p})$ and $\boldsymbol{\phi}^{\dagger}(\mathbf{p})$. For the Bose-Einstein case, show that the mixed commutator of field operators in coordinate and momentum space is
$$
\left[\boldsymbol{\phi}(\mathbf{p}), \boldsymbol{\psi}^{\dagger}(\mathbf{r})\right]=\langle\mathbf{p} \mid \mathbf{r}\rangle
$$

Quantum mechanics

Eugen Merzbacher 3rd Edition

Chapter 21, Problem 4

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Step 1

Step 1: **Define the momentum space annihilation operator** The momentum space annihilation operator $\boldsymbol{\phi}(\mathbf{p})$ is defined as: $$ \boldsymbol{\phi}(\mathbf{p}) = \int \langle \mathbf{p} | \mathbf{r} \rangle \boldsymbol{\psi}(\mathbf{r}) d^3 r Show more…

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Defining the momentum space annihilation operator ${ }^3$ $$ \boldsymbol{\phi}(\mathbf{p})=\int\langle\mathbf{p} \mid \mathbf{r}\rangle \boldsymbol{\psi}(\mathbf{r}) d^3 r $$ derive the commutation (or anticommutation) relations for $\boldsymbol{\phi}(\mathbf{p})$ and $\boldsymbol{\phi}^{\dagger}(\mathbf{p})$. For the Bose-Einstein case, show that the mixed commutator of field operators in coordinate and momentum space is $$ \left[\boldsymbol{\phi}(\mathbf{p}), \boldsymbol{\psi}^{\dagger}(\mathbf{r})\right]=\langle\mathbf{p} \mid \mathbf{r}\rangle $$

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