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Q3

Part of STEMS 2021 Phy Cat C

Problems(1)

STEMS 2021 Phy Cat C Q3

Source:

1/23/2021
Classical Probability Distribution for Quantum States? The goal of this problem is to try and mimic a Statistical Mechanics approach to Quantum Mechanics. In Classical Statistical Mechanics one has the usual Gibbs-Boltzmann Formula which gives the probability distribution in phase-space to be: ρ(x1,…,xn,p1,…,pn)∼exp⁡(−βH(x1,…,xn,p1,…,pn)) \rho(x_1,\dots,x_n,p_1,\dots,p_n) \sim \exp(-\beta H(x_1,\dots,x_n,p_1,\dots,p_n)) where HH is the Hamiltonian of the system.
[*] Why can't we demand a similar probability distribution over phase-space in Quantum Mechanics? \\ If the wave function ψ(x1,…,xn)\psi(x_1,\dots,x_n) is given, we construct the following expression: \begin{align*} \begin{split} & P(x_1,\dots,x_n,p_1,\dots,p_n) \\ & = \left(\frac{1}{\pi\hbar}\right)^n \int_{-\infty}^{\infty} \dots \int_{-\infty}^{\infty} dy_1\dots dy_n \psi^*(x_1+y_1,\dots,x_n+y_n) \\ & \times \psi(x_1-y_1,\dots,x_n-y_n) \exp\left(\frac{2i}{\hbar}(p_1y_1+\dots+p_ny_n)\right) \end{split} \end{align*}[/*] [*] Show that, ∫−∞∞ ⁣⋯∫−∞∞dp1…dpnP(x1,…,xn,p1,…,pn)=∣ψ(x1,…,xn)∣2 \int_{-\infty}^{\infty}\dots\int_{-\infty}^{\infty} dp_1\dots dp_n P(x_1,\dots,x_n,p_1,\dots,p_n) = \left|\psi(x_1,\dots,x_n)\right|^2 which are the correct probabilities for the co-ordinates. [/*]
[*] Show that, ∫−∞∞ ⁣⋯∫−∞∞dx1…dxn P(x1,…,xn,p1,…,pn)=∣ψ~(p1,…,pn)∣2 \int_{-\infty}^{\infty}\dots\int_{-\infty}^{\infty} dx_1\dots dx_n \, P(x_1,\dots,x_n,p_1,\dots,p_n) = \left|\tilde{\psi}(p_1,\dots,p_n)\right|^2 which are the correct probabilities for the momenta where, ψ~(p1,…,pn)=∫−∞∞ ⁣⋯∫−∞∞dx1…dxnψ(x1,…,xn)exp⁡(−iℏ(x1p1+⋯+xnpn)) \tilde{\psi}(p_1,\dots,p_n) = \int_{-\infty}^{\infty}\dots\int_{-\infty}^{\infty} dx_1\dots dx_n \psi(x_1,\dots,x_n) \exp\left(-\frac{i}{\hbar}(x_1p_1+\dots+x_np_n)\right) is the Fourier transform of the wave-function ψ(x1,…,xn)\psi(x_1,\dots,x_n). [/*]
[*] The function PP defined above therefore seems to be a good candidate for a probability distribution in Quantum Mechanics. Would this not contradict part (a)? Give reasons to support your answer. [/*]