Foundations of Quantum Mechanics (ACM30210) Set 3 Dr Lennon O´ N´araigh 20th February 2020 Instructions: This is a graded assignment. • Put your name and student number on your homework. • The assignment is to be handed in on Monday March 9th before 13:00. • Hand in your assignment by putting it into the homework box marked ‘ACM 30210’ • outside the school office (G03, Science North). Model answers will be made available online in due course. • For environmental reasons, please do not place your homework in a ‘polly pocket’ • – use a staple instead. Preliminaries In these exercises, let ψ(x, t) be the wavefunction of for a single-particle system satisfying Schr¨odinger’s equation, ∂ψ ~2 i~ = 2 + U(x) ψ, (1) ∂t −2m∇ where all the symbols have their usual meaning. 1 Quantum Mechanics Madelung Transformations 1. Write ψ in ‘polar-coordinate form’, ψ(x, t) = R(x, t)eiθ(x,t), where R and θ are both functions of space and time. Show that: ∂R ~ + R2 θ =0. (2) ∂t 2mR∇· ∇ Hint: Use the ‘conservation law of probability’ in Chapter 9 of the lecture notes. 2. Define 2 R = ρm/m mR = ρm, (3a) ⇐⇒ and: p 1 θ = S/~ S = θ~, u = S. (3b) ⇐⇒ m∇ Show that Equation (4) can be re-written as: ∂ρm + (uρm)=0. (4) ∂t ∇· 3. Again using the representation ψ = Reiθ, show that S = θ~ satisfies the Quantum Hamilton–Jacobi equation: ∂S 1 ~2 2R + S 2 + U = ∇ . (5) ∂t 2m|∇ | 2m R 4. Starting with Equation (5), deduce: ∂u 1 + u u = UQ, (6) ∂t ·∇ −m∇ where UQ is the Quantum Potential, 2 2 ~ √ρm UQ = U ∇ . − 2m √ρm 5. Argue (using only a few lines and the postulates of Quantum Mechanics) that the circulation Γ= mu dr satisfies Γ=2nπ~, where n is an integer. C · R3 Here, C is any closedH path in and the line integral C mu dr is defined in the usual way as in Vector Calculus. · H In the next and final exercises, throw away your Quantum Mechanics knowledge for a second and cast your mind back to Classical Mechanics, where a particle experiencing a potential U(x) satisfies Newton’s Law: dp dx = U, p = m . dt −∇ dt Thus, a physical system in Classical Mechanics is described by a trajectory (x(t), p(t)) in the phase space R6. Here, the trajectory should properly be written as: (x(t; x0, p0), p(t, x0, p0)), to indicate the dependence on the initial conditions x0 and p0. 2 Quantum Mechanics Madelung Transformations 6. Show that Newton’s Law can be re-written as dx ∂H dp ∂H i = , i = , dt ∂pi dt −∂xi where the index i = 1, 2, 3 labels the x-, y-, and z-directions of physical space. Here also, H is the classical Hamiltonian, p2 H = + U(x). 2m 7. Hence or otherwise, show that Newton’s Law is equivalent to: 3 ∂S 1 2 + xS + U(x)=0, S = pidxi Et, (7) ∂t 2m |∇ | − i=1 ! X Z where E is the constant energy of the system and H is the Classical Hamiltonian. 3 Here, S = pidxi Et is called the Classical Action. i=1 − P R Interpretation – Madelung Transformation The equations ∂ρm + (uρm) = 0, (8a) ∂t ∇· 2 2 ∂u 1 ~ √ρm + u u = UQ, UQ = U ∇ . (8b) ∂t ·∇ −m∇ − 2m √ρm provide the Madelung form of Schr¨odinger’s equation. The transformations that take us from Schr¨odinger’s equation to Equations (8) are called the Madelung transfor- mation. Equations (8) have exactly the same form as the barotropic Euler equations of Fluid Mechanics. Thus, on a purely formal level, Quantum Mechanics for particles is equivalent to Classical Fluid Mechanics for inviscid flows. The quantum part of the 2 2 potential (~ /2m)( √ρm)/√ρm can be identified with the pressure. The pressure therefore− depends on∇ the density through a highly specialized equation of state – ef- fectively a barotropic fluid. The Madelung formulation of Quantum Mechanics is very useful for modelling many-body systems involving both nuclei which behave classically and electrons which behave as described in Quantum Mechanics. Caution! There is a reason the Madelung equations are not usually introduced in intro- ductory courses on Quantum Mechanics – they are non-linear, whereas the transformed Schr¨odinger equation is linear. It is only for the very unusual barotropic fluid with the ‘quantum pressure’ that the non-linear Euler equation can be transformed into a lin- ear Schr¨odinger eqeuation. Such a transformatoin cannot be found for a general fluid. The transformation is similar to the Cole–Hopf transformation which maps the nonlinear Burgers equation of PDE theory to the Heat Equation – again, this is a very special transformation for a very special equation (Burgers). 3 Quantum Mechanics Madelung Transformations Interpretation – Hamilton–Jacobi Equation (HJE) The Classical HJE can be thought of as the solution (to lowest order in perturbation theory) to the Quantum HJE. The pertinent perturbation theory is a singular perturba- tion theory in the small parameter ~. Thus, the Quantum HJE (5) and the Classical HJE (7) and the are in form identical to lowest order in ~. Thus, the passage from Classical Mechanics to Quantum Mechanics can be made by identifying the classical action function S θ~, where θ is the phase of the wavefunction. → 4.