Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
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Question
Chapter 10, Problem 10.18P
To determine
Compute the transmission coefficient for scattering from a delta function and from a finite square well and compare the delta function transmission coefficient with equation 2.144 and finite square well potential transmission coefficient with equation 2.172.
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1 W:0E
*Problem 1.3 Consider the gaussian distribution
p(x) = Ae¬^(x-a)²
%3D
where A, a, and A are positive real constants. (Look up any integrals you need.)
(a) Use Equation 1.16 to determine A.
(b) Find (x), (x²), and ơ.
(c) Sketch the graph of p(x).
Show that a gaussian psi (x) = e ^(-ax^2) can be an eigenfunction of H(hat) for harmonic oscillator
1. Compute T(hat)*psi
2. Compute Vhat* psi - assume V operator is 1/2w^2x^2
3. Write out Hbar*psi and identify terms so Hber*psi=E*psi is true
4. From cancellation find a
5. insert back a to Schrodinger eq above and find E
Problem 2.11
(a) Compute (x). (p). (x²), and (p²), for the states yo (Equation 2.60) and 1 (Equation
2.63), by explicit integration. Comment: In this and other problems involving the
harmonic oscillator it simplifies matters if you introduce the variable = √mo/hx
and the constanta (m/h)¹/4
(b) Check the uncertainty principle for these states.
(c) Compute (T) and (V) for these states. (No new integration allowed!) Is their sum
what you would expect?
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- A pendulum hangs from a fixed point, but the pendulum itself is a spring with a constantk and length I Assume that the whole thing moves in the vertical plane. How many connections are there and how many generalized coordinates? Give the Lagrangian the equation of motion. Calculate the equilibrium position. Argue why it is a stable equilibrium. Calculate the normal vibration frequency.arrow_forwardConsider the length of the simple pendulum is 1 and the mass of the pendulum bob is m. Obtain the equation for the Lagrangian for the simple pendulum when its kinetic energy is 1 mlo and the potential energy is mgl(1–cos ø) . m². 2. 1 (A) -– mgl (1- cos ø) ml²ø² – 1. (B) – ml o-mgl(1-coso) 2 1 (C)ml'o +mgl(1-cos o) 2. 1 (D) ml o-2mgl (1- cos o) Answer Activate W Go to Settings D. A,arrow_forwardCalculate the values of ⟨x3⟩v and ⟨x4⟩v for a harmonic oscillator by using the properties of the Hermite polynomials given in Table 7E.1; follow the approach used in the text.arrow_forward
- Determine the transmission coefficient for a rectangular barrier (same as Equation 2.127, only with +Vo in the region -a Vo (note that the wave function inside the barrier is different in the three cases). Partial answer: For Earrow_forwardProblem 2.14 In the ground state of the harmonic oscillator, what is the probability (correct to three significant digits) of finding the particle outside the classically allowed region? Hint: Classically, the energy of an oscillator is E = (1/2) ka² = (1/2) mo²a², where a is the amplitude. So the “classically allowed region" for an oscillator of energy E extends from –/2E/mw² to +/2E/mo². Look in a math table under “Normal Distribution" or "Error Function" for the numerical value of the integral, or evaluate it by computer.arrow_forwardProblem 2.7 A particle in the infinite square well has the initial wave function JAx, У (х, 0) — 0< x < a/2, a/2 < x < a. А (а — х), (a) Sketch ¥ (x, 0), and determine the constant A. (b) Find ¥(x, t). (c) What is the probability that a measurement of the energy would yield the value E1? (d) Find the expectation value of the energy, using Equation 2.21.21arrow_forwardFor Problem 8.16, how do I prove the relations and give the correct expressions?arrow_forwardHow would I be able to sketch the graph in problem 7.36?arrow_forwardA prticle in the infinite square wen has its inidal wave fumtionan even mixture of the first two stationary states: Normalize IV(x. 0).Find IV(x, t) and IV(x,0)I2 Compute<x>. Notice that it oscillates in time. What is the angular frequencyof the oscillation? What is the amplitude of the oscillation? Compute <p>.If you measured the energy what values might you gets andwhat is the probability the valueOf H .HOW does it compare with E1and E2 *** Please pay attention to the picture***arrow_forwardSolve the time-independent Schrödinger equation with appropriate boundary conditions for an infinite square well centered at the origin [V (x) = 0, for -a/2 < x < +a/2; V (x) = 00 otherwise]. Check that your allowed energies are consistent with mine (Equation 2.23), and confirm that your y's can be obtained from mine (Equation 2.24) by the substitution x x - a/2.arrow_forward(a) Write down the wave functions for the three regions of the potential energy barrier (Figure 5.25) for E < U₁. You will need six coefficients in all. Use complex exponential notation. (b) Use the boundary conditions at x = 0 and at x = L to find four relationships among the six coeffi- cients. (Do not try to solve these relationships.) (c) Sup- pose particles are incident on the barrier from the left. Which coefficient should be set to zero? Why?arrow_forward6.2 Let the "uniform" ensemble of energy E be defined as the ensemble of all systems of the given type with energy less than E. The equivalence between (6.29) and (6.27) means that we should obtain the same thermodynamic functions from the "uniform" ensemble of energy E as from the microcanonical ensemble of energy E. In particular, the internal energy is E in both ensembles. Explain why this seemingly paradoxical result is true.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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