* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Relativity Problem Set 9
Second quantization wikipedia , lookup
Scalar field theory wikipedia , lookup
EPR paradox wikipedia , lookup
Hidden variable theory wikipedia , lookup
Coherent states wikipedia , lookup
Hydrogen atom wikipedia , lookup
Dirac equation wikipedia , lookup
Quantum state wikipedia , lookup
Wheeler's delayed choice experiment wikipedia , lookup
Renormalization wikipedia , lookup
Coupled cluster wikipedia , lookup
Elementary particle wikipedia , lookup
Ensemble interpretation wikipedia , lookup
Canonical quantization wikipedia , lookup
Molecular Hamiltonian wikipedia , lookup
Path integral formulation wikipedia , lookup
Identical particles wikipedia , lookup
Tight binding wikipedia , lookup
Schrödinger equation wikipedia , lookup
Atomic theory wikipedia , lookup
Introduction to gauge theory wikipedia , lookup
Probability amplitude wikipedia , lookup
Aharonov–Bohm effect wikipedia , lookup
Renormalization group wikipedia , lookup
Double-slit experiment wikipedia , lookup
Bohr–Einstein debates wikipedia , lookup
Copenhagen interpretation wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Particle in a box wikipedia , lookup
Wave function wikipedia , lookup
Wave–particle duality wikipedia , lookup
Matter wave wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Relativity Problem Set 9 Prof. J. Gerton Due Tuesday November 8, 2011 at the beginning of class Problem 1 (10 pts.) The quantum harmonic oscillator A Quantum Harmonic Oscillator (QHO) in one dimension describes a quantum particle moving in the harmonic potential V (x) = mω 2 2 x. 2 The energy levels are quantized according to 1 En = ~ ω n + , 2 (1) (2) where n = 0, 1, 2, ... is an integer quantum number. (a) The wave function for the QHO in its ground state (n = 0) is a Gaussian, of the form 2 ψ(x) = A e−B x . (3) Using the Schroedinger equation with the harmonic potential V (x) and the ground state energy E0 , find the constant B. (b) Find the constant A by imposing the normalization of the wave function. Problem 2 (10 pts.) Expectation values for a harmonic oscillator Consider the wave function ψ1 (x), describing the first excited state of a QHO with frequency ω. (a) Compute the expectation value of the momentum of a QHO in the first excited state, that is, compute Z +∞ (4) hpi = dx ψ1∗ (x) p̂ ψ1 (x), −∞ where a star (*) indicates a complex conjugation. Notice that the ordering of the factors inside the integral matters! 1 2 (b) Compute the expectation value of the position operator x̂ in the first excited state. (c) Compute the expectation value of the kinetic energy operator K̂ in the first excited state. Problem 3 (10 pts.) The Evanescent Wave A beam of free (unbound) particles, all with mass m, is incident from the left (x = −∞) on a step-up potential of the form ( 0, for x < 0, V (x) = (5) V0 , for x ≥ 0. We now consider the case where the total energy of each particle is smaller than the potential height, E < V0 . (a) Write down the wave function ψ(x) in the region x > 0. (b) Recall that for a beam of free particles, ψ ∗ (x)ψ(x) gives the number of particles per unit distance. Using this, discuss whether it would be possible to find a particle in the region x > 0 if a measurement were made on the system. (c) What is the probability that an incident particle will be reflected by the potential barrier? (d) Are your answers to (b) and (c) contradictory? Think hard and offer a tentative explanation for this. Problem 4 (10 pts.) The double well - Part 1 We consider the potential +∞ for x < −a and for x > a; V (x) = 0 for −a ≤ x < −b and for b < x ≤ a V0 for −b ≤ x ≤ b; (6) Consider a particle of mass m and energy E < V0 moving in such potential. (a) Draw the potential V (x). (b) What is the wave function for x > a and x < −a? course name PS # 3 (c) Using the conservation of energy relation, find the momentum of the particle in all regions of the well. Indicate with p the momentum in the regions [−a, −b] and [a, b], while you indicate the momentum in the region [−b, b] as iκ (yes, the momentum is imaginary in this region!). (d) Can the particle be in the region [−b, b] in classical mechanics? What happens here? Problem 5 (10 pts.) The double well - Part 2 We refer to Problem 4 above. (a) Write down the wave functions for the particle in the various regions of the double well potential. Remember that in any region, the wave function is of the form ψ(x) = C1 e−ik̃i x + C2 eik̃x , (7) where k̃i is the momentum in the i−th region, given by the solution to (c) above, and C1,i and C2,i are arbitrary constants for the region i. In particular, what is the wave function in the region [−b, b]? Notice that in the definition of the wave function you should have six coefficients total. (b) Impose the symmetry of the wave function, that is, impose ψ(−x) = ψ(x). (8) This will reduce the number of the coefficients from six to three. What is the wave function in [−b, b] after the symmetrization? (c) Impose the continuity of the wave function everywhere, and the continuity of its derivative at x = −b (or at x = b, which is the same thanks to the symmetrization). From these equations, you should derive the expression p = k tanh(κb) tan(p(b − a)), (9) where tanh(x) is the hyperbolic tangent and tan(x) is the usual tangent. (d) Write the equation above in terms of E, V0 , m, a and b. course name PS #