MathDB

STEMS 2021 Phy Cat C

Part of 2021 India STEMS

Subcontests

(3)
Q3
1

STEMS 2021 Phy Cat C Q3

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. [/*]
Q2
1
Q1
1

STEMS 2021 Phy Cat C Q1

Black hole thermodynamics The goal of this problem is to explore some interesting properties of Black Holes. The following equation was obtained by L. Smarr in 1973: M2=116πA+4πA(J2+14Q4)+12Q2 M^2 = \frac{1}{16\pi} A + \frac{4\pi}{A}\left(J^2 + \frac{1}{4}Q^4\right)+\frac{1}{2}Q^2 where MM, JJ, QQ and AA are the mass, angular momentum, charge and area of the event horizon of a black hole.
To make contact with thermodynamics we write for the entropy of the Black Hole, S=14kBAS = \frac{1}{4}k_B A where kBk_B is the Boltzmann constant.
[*] Work in natural units G=ℏ=c=1G = \hbar = c = 1 and show that the equation for the entropy is dimensionally correct. [/*] [*] Take kB=1/8πk_B = 1/8\pi (by choosing units) and derive an expression for S(M,J,Q)S(M,J,Q). Is this expression unique? (Hint: What is the entropy of the Schwarzschild Black Hole which corresponds to J=Q=0J=Q=0?) \item We suppose the mass-energy MM (since c=1c=1) plays the role of internal energy. Show that T,Ω,ΦT,\Omega,\Phi defined via, dM=TdS+ΩdJ+ΦdQ dM = T dS + \Omega dJ + \Phi dQ are given by, \begin{eqnarray*} & T = \frac{1}{M} \left[1- \frac{1}{16S^2}\left(J^2 + \frac{1}{4}Q^4\right)\right] \\ & \Omega = \frac{J}{8MS}\\ & \Phi = \frac{Q}{2M}\left[1+\frac{Q^2}{8S}\right]. \end{eqnarray*} This is the analog of the first law of thermodynamics. [/*] [*]Look at the expression for M(S,J,Q)M(S,J,Q) closely and derive the analog of the Gibbs-Duhem Relation familiar from Thermodynamics. [/*] [*] Show that, S→14M2−18Q2 S \to \frac{1}{4}M^2 - \frac{1}{8}Q^2 as T→0T \to 0. What does this say about the third law of thermodynamics? Give reasons to support your answer. \item An alternative statement to the third law is that "it is impossible to reach absolute-zero in a finite number of steps". What can we conclude from part (e)? [/*]