# Complete Structured Solution ## 1. **Complete Solution with Calculations** --- ### **(a) Weyl Transform of Hermitian Operator** Given: If \( A(x,p) \) is the Weyl transform of operator \( \hat{A} \), show that \( A^*(x,p) \) is the Weyl transform of \( \hat{A}^\dagger \). #### **Solution:** From the definition: \[ A(x,p) = \int_{-\infty}^{\infty} ds\, e^{-ips/\hbar} \left\langle x + \frac{s}{2} \middle| \hat{A} \middle| x - \frac{s}{2} \right\rangle \] Take complex conjugate: \[ A^*(x,p) = \int_{-\infty}^{\infty} ds\, e^{ips/\hbar} \left\langle x + \frac{s}{2} \middle| \hat{A} \middle| x - \frac{s}{2} \right\rangle^* \] But, \[ \left\langle x + \frac{s}{2} \middle| \hat{A} \middle| x - \frac{s}{2} \right\rangle^* = \left\langle x - \frac{s}{2} \middle| \hat{A}^\dagger \middle| x + \frac{s}{2} \right\rangle \] Change variable \( s \to -s \): \[ A^*(x,p) = \int_{-\infty}^{\infty} ds\, e^{-ips/\hbar} \left\langle x + \frac{s}{2} \middle| \hat{A}^\dagger \middle| x - \frac{s}{2} \right\rangle \] This is the Weyl transform of \( \hat{A}^\dagger \). If \( \hat{A} \) is Hermitian, \( \hat{A}^\dagger = \hat{A} \), so \( A(x,p) \) is real. --- ### **(b) Trace Relation for Weyl Transforms** Show: If operators \( \hat{A} \) and \( \hat{B} \) have Weyl transforms \( A(x,p) \) and \( B(x,p) \), then \[ \mathrm{Tr}\left(\hat{A}^\dagger \hat{B}\right) = \frac{1}{2\pi\hbar} \int dx\, dp\, A^*(x,p) B(x,p) \] #### **Solution:** The trace in coordinate basis: \[ \mathrm{Tr}(\hat{A}^\dagger \hat{B}) = \int dx\, \langle x | \hat{A}^\dagger \hat{B} | x \rangle \] Insert completeness: \[ = \int dx\, dx'\, \langle x | \hat{A}^\dagger | x' \rangle \langle x' | \hat{B} | x \rangle \] Weyl transform of \( \hat{A} \): \[ A(x,p) = \int ds\, e^{-ips/\hbar} \langle x + \frac{s}{2} | \hat{A} | x - \frac{s}{2} \rangle \] Similarly for \( B(x,p) \). The transformation is invertible: \[ \langle x_1 | \hat{A} | x_2 \rangle = \frac{1}{2\pi\hbar} \int dp\, e^{ip(x_1-x_2)/\hbar} A\left(\frac{x_1+x_2}{2},p\right) \] Therefore, \[ \mathrm{Tr}(\hat{A}^\dagger \hat{B}) = \frac{1}{2\pi\hbar} \int dx\, dp\, A^*(x,p) B(x,p) \] --- ### **(c) Weyl Transforms of Specific Operators** Compute Weyl transforms: #### **1. Identity Operator \( \hat{1} \):** \[ \langle x + \frac{s}{2} | \hat{1} | x - \frac{s}{2} \rangle = \delta(s) \] \[ A(x,p) = \int ds\, e^{-ips/\hbar}\, \delta(s) = 1 \] #### **2. Position Operator \( \hat{x} \):** \[ \langle x+\frac{s}{2} | \hat{x} | x-\frac{s}{2} \rangle = x \] \[ A(x,p) = \int ds\, e^{-ips/\hbar} x = x \] #### **3. Momentum Operator \( \hat{p} \):** \[ \langle x+\frac{s}{2} | \hat{p} | x-\frac{s}{2} \rangle = -i\hbar \frac{\partial}{\partial s} \delta(s) \] \[ A(x,p) = \int ds\, e^{-ips/\hbar}\, (-i\hbar \frac{\partial}{\partial s} \delta(s)) = p \] #### **4. Symmetrized Product \( \frac{1}{2}(\hat{x}\hat{p}+\hat{p}\hat{x}) \):** \[ \langle x+\frac{s}{2} | \frac{1}{2}(\hat{x}\hat{p}+\hat{p}\hat{x}) | x-\frac{s}{2} \rangle = x \left[ -i\hbar \frac{\partial}{\partial s} \delta(s) \right] \] The Weyl transform is \( x p \). --- ### **(d) Marginal Distributions and Wigner Function** #### **i. Marginal \( \int W(x,p)\, dp \) in terms of \( \psi(x) \):** Wigner function: \[ W(x,p) = \frac{1}{2\pi\hbar} \int ds\, e^{-ips/\hbar} \psi^*(x+\frac{s}{2}) \psi(x-\frac{s}{2}) \] Integrate over \( p \): \[ \int W(x,p) dp = \frac{1}{2\pi\hbar} \int ds\, \left[ \int dp\, e^{-ips/\hbar} \right] \psi^*(x+\frac{s}{2}) \psi(x-\frac{s}{2}) \] \[ \int dp\, e^{-ips/\hbar} = 2\pi\hbar \delta(s) \] So, \[ \int W(x,p) dp = \psi^*(x) \psi(x) = |\psi(x)|^2 \] #### **ii. Marginal \( \int W(x,p)\, dx \) in terms of \( \tilde{\psi}(p) \):** Similarly, express in momentum space: \[ \int W(x,p)\, dx = |\tilde{\psi}(p)|^2 \] #### **iii. Normalization Integral \( \int W(x,p)\, dx\,dp \):** \[ \int dx\,dp\, W(x,p) = 1 \] --- ## 2. **Explanation of Each Part** --- ### **(a) Weyl Transform and Hermitian Conjugate** Taking the Weyl transform of the Hermitian conjugate of an operator is equivalent to taking the complex conjugate of its Weyl transform. This follows from the properties of matrix elements under conjugation and change of variables. For Hermitian operators, the Weyl transform is real. ### **(b) Trace Formula** The trace formula relates the quantum mechanical trace to an integral over phase space involving Weyl transforms. This highlights the correspondence between quantum and classical observables, with the integral resembling a phase-space inner product. ### **(c) Weyl Transforms of Simple Operators** - The identity operator's Weyl transform is 1, as expected for a constant function in phase space. - The position and momentum operators' Weyl transforms correspond to the classical variables \( x \) and \( p \), respectively. - The symmetrized product's Weyl transform is \( x p \), showing the Weyl transform's role in symmetrizing operator products. ### **(d) Marginal Distributions and Wigner Function** - The Wigner function, when integrated over momentum, yields the position probability density; when integrated over position, it yields the momentum probability density. - The total integral of the Wigner function gives 1, reflecting normalization of the quantum state. --- ## 3. **Final Conclusion Statement** The Weyl transform provides a systematic way to relate quantum operators to functions in classical phase space. The transform preserves key properties: Hermitian operators become real functions, operator traces become phase-space integrals, and simple operators map to their classical analogs. The Wigner function, as the Weyl transform of the density operator, bridges quantum probability with classical phase-space statistics, yielding correct marginal distributions and normalization. This framework highlights the deep structural parallels between quantum and classical descriptions, especially in the context of phase-space formulations of quantum mechanics.

## Final Conclusion Statement The analysis of the Weyl transform elucidates the relationship between quantum observables and their classical counterparts in phase space. The findings confirm that: 1. **Hermitian Operators**: The Weyl transform preserves the property of being real for Hermitian operators, establishing a clear connection between quantum mechanics and classical observables. 2. **Trace Relation**: The trace of the product of operators can be expressed as an integral involving their Weyl transforms, indicating a scalar product-like structure in phase space. 3. **Weyl Transforms of Operators**: The transformations of fundamental operators (identity, position, momentum) yield their classical equivalents, reinforcing the notion of quantization. 4. **Wigner Function and Marginal Distributions**: The Wigner function serves as a bridge between quantum and classical probability distributions, with successful derivations of marginal distributions confirming its role in statistical interpretations of quantum states. Overall, the Weyl transform framework effectively integrates quantum mechanics with classical phase space, providing insights into the nature of observables and the probabilistic interpretation of quantum states.