8. Hamilton's equations of motion are invariant under a class of transformations that mix up coordinates and momenta, known as ""canonical transformations."" A more advanced (i.e. graduate level) course would treat this in depth, but here let's just look at a curious example. a) Show that the Hamiltonian of a 1-D harmonic oscillator with unit mass and unit force constant is $H = frac{1}{2}(q^2 + p^2)$ b) Consider this canonical transformation to new canonical coordinates ($Q$, $P$): $q = sqrt{2P} sin Q$ $p = sqrt{2P} cos Q$ Show that if $q$ and $p$ satisfy Hamilton's equations 13.17, then $Q$ and $P$ satisfy the same equations. c) Rewrite the Hamiltonian of part (a) in terms of the new coordinates, by substitution of the transformation equations, and show that $Q$ is an ignorable coordinate. d) From looking at the transformed Hamiltonian, what is the canonical momentum $P$? e) Solve for $Q(t)$, and by substituting the transformation equations, show that the result for $q(t)$ is the expected solution to the harmonic oscillator problem. There exists a systematic way of finding such transformations that result in the new coordinate(s) being ignorable, by solving a partial differential equation called the Hamilton- Jacobi equation. Interestingly, that equation is an analog to the Schrödinger equation in quantum mechanics and can be derived from the quantum-mechanics equation in the limit that the Planck constant goes to zero.
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For a harmonic oscillator, the potential energy is given by: V = (1/2)kx^2 where k is the force constant and x is the displacement from the equilibrium position. Since the mass is unit, the kinetic energy is given by: T = (1/2)m(v^2) = (1/2)(1)(v^2) = Show more…
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2. Schrodinger equation In quantum mechanics, physical quantities correspond to Hermitian operators. In particular, the total energy of the system corresponds to the Hamiltonian operator H, which is a hermitian operator. The ‘state of the system’ is a time dependent vector in an inner product space, |Έ(t)⌡. The state of the system obeys the Schrodinger equation iħ d/dt |Έ(t)⌡ = H|Έ(t)⌡. We assume that there are no time-varying external forces on the system, so that the Hamiltonian operator H is not itself time-dependent. a) Take the Hermitian conjugate of the Schrodinger equation, which puts it into bra form (this is a simple one-liner; use the fact that H is hermitian). b) Show that 〈Έ(t)|Έ(t)⌡ is time independent, i.e. d/dt 〈Έ(t)|Έ(t)⌡ = 0. (hint: use the product rule). This means that we can normalize the state vector at one time, say t = 0, so that 〈Έ(0)|Έ(0)⌡ = 1, and it will stay normalized at all times, 〈Έ(t)|Έ(t)⌡ = 1. c) Define the time-evolution operator U(t) = e^{-iHt/ħ} where the exponential is defined by a Taylor expansion in powers of H: U(t) = I - (it/ħ)H + 1/2 (-it/ħ)^2 H^2 + ... Show that U(t) is unitary: U(t)† U(t) = U(t)U(t)† = I. d) Let f(H) be any function of H that can be defined by a Taylor expansion in powers of H. Show that H and f(H) commute. e) Find an expression for d/dt U(t). f) Show that |Έ(t)⌡ = U(t)|Έ(0)⌡ is a solution of the Schrodinger equation. Here |Έ(0)⌡ is the initial state of the system, which we assume is given.
Madhur L.
(a) Write down the one-dimensional time-dependent Schrödinger equation for the wavefunction Ψ(x,t) when the potential energy is V(x). Using a wavefunction of the form Ψ(x,t) = ψ(x) e^(-iEt/ħ), show that this 1D equation can be transformed into the time-independent Schrödinger equation (TISE). Explain clearly why solutions of this form are called stationary states. (b) Use the Heisenberg momentum-position uncertainty relation to estimate the momentum and the kinetic energy of a particle trapped in a one-dimensional box of width L. The potential inside the box is zero, and infinite outside. (c) Find the eigenvalues and the associated normalised eigenvectors of the spin-1/2 matrix Sy = (ħ/2)σy, where σy = (0 -i; +i 0). Comment on the eigenvalues obtained. (d) Consider a system of 4 non-interacting fermions with spin 1/2 (two with spin up and two with spin down) in an infinite square well. The energy levels available for each of the particles are En = n^2 E1, where n = 1, 2, 3... and E1 is the energy of the lowest available level. i. What is the total energy of the system of 4 particles in its ground state? Give also the total energy and configuration of the system when in its 2nd excited state. ii. What would the ground state energy of the system be if all 4 particles were bosons rather than fermions? Give reasons for all your answers above. (e) Find the commutator [p̂x, x̂²], using the fact that p̂x = -iħ(∂/∂x) and x̂ is the usual position operator.
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Example 1.2: Simple Harmonic Oscillator using Complex Variables The above phase space real variables are coupled as can be seen via (1.1.14) and (1.1.15). We can transform to complex variables in order to dis-entangle this system as follows H = 1/2 (p^2 + ω^2q^2) = 1/2 (ωq + ip)(ωq - ip) a = 1/sqrt(2ω) (ωq + ip), (1.1.18) a* = 1/sqrt(2ω) (ωq - ip), such that the Hamiltonian is given by H = ωa*a = ω|a|^2 (1.1.19) with the algebraic relation {a(q, p), a*(q, p)} = -i. (1.1.20) These equations are very similar to the ones seen in the quantized version of this system. Note that this Hamiltonian is still real even though we have used complex variables. Observables in classical mechanics must be real functions (just are observables in quantum mechanics are self adjoint operators). Using this Hamiltonian one can show, using the Leibniz rule and (1.1.20), that da/dt = {a, H} = {a, ωa*a} = ω{a, a*}a + ωa*{a, a} = -iωa (1.1.21) with a simple solution of a(t) = a(0)e^-iωt. Since the variables (1.1.18) are complex conjugates with each other, one has a*(t) = a*(0)e^iωt.
Sri K.
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