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 2, Problem 2.47P
(a)
To determine
The ground state and the first excited state wave functions for the given case.
(b)
To determine
The variation in energies with respect to
(c)
To determine
Whether the electron draw the nuclei together or push them apart.
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2.4.
A particle moves in an infinite cubic potential well described by:
V (x1, x2) = {00
12=
if 0 ≤ x1, x2 a
otherwise
1/2(+1)
(a) Write down the exact energy and wave-function of the ground state.
(2)
(b) Write down the exact energy and wavefunction of the first excited states and specify their
degeneracies.
Now add the following perturbation to the infinite cubic well:
H' = 18(x₁-x2)
(c) Calculate the ground state energy to the first order correction.
(5)
(d) Calculate the energy of the first order correction to the first excited degenerated state.
(3)
(e) Calculate the energy of the first order correction to the second non-degenerate excited
state.
(3)
(f) Use degenerate perturbation theory to determine the first-order correction to the two initially
degenerate eigenvalues (energies).
(3)
Problem 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.
PROBLEM 2. Consider a spherical potential well of radius R and depth Uo,
so that the potential is U(r) = -Uo at r R.
Calculate the minimum value of Uc for which the well can trap a particle
with l = 0. This means that SE at Uo > Uc has at least one bound ground
state at l = 0 and E < 0. At Ug = Uc the bound state disappears.
Chapter 2 Solutions
Introduction To Quantum Mechanics
Ch. 2.1 - Prob. 2.1PCh. 2.1 - Prob. 2.2PCh. 2.2 - Prob. 2.3PCh. 2.2 - Prob. 2.4PCh. 2.2 - Prob. 2.5PCh. 2.2 - Prob. 2.6PCh. 2.2 - Prob. 2.7PCh. 2.2 - Prob. 2.8PCh. 2.2 - Prob. 2.9PCh. 2.3 - Prob. 2.10P
Ch. 2.3 - Prob. 2.11PCh. 2.3 - Prob. 2.12PCh. 2.3 - Prob. 2.13PCh. 2.3 - Prob. 2.14PCh. 2.3 - Prob. 2.15PCh. 2.3 - Prob. 2.16PCh. 2.4 - Prob. 2.17PCh. 2.4 - Prob. 2.18PCh. 2.4 - Prob. 2.19PCh. 2.4 - Prob. 2.20PCh. 2.4 - Prob. 2.21PCh. 2.5 - Prob. 2.22PCh. 2.5 - Prob. 2.23PCh. 2.5 - Prob. 2.24PCh. 2.5 - Prob. 2.25PCh. 2.5 - Prob. 2.26PCh. 2.5 - Prob. 2.27PCh. 2.5 - Prob. 2.28PCh. 2.6 - Prob. 2.29PCh. 2.6 - Prob. 2.30PCh. 2.6 - Prob. 2.31PCh. 2.6 - Prob. 2.32PCh. 2.6 - Prob. 2.34PCh. 2.6 - Prob. 2.35PCh. 2 - Prob. 2.36PCh. 2 - Prob. 2.37PCh. 2 - Prob. 2.38PCh. 2 - Prob. 2.39PCh. 2 - Prob. 2.40PCh. 2 - Prob. 2.41PCh. 2 - Prob. 2.42PCh. 2 - Prob. 2.44PCh. 2 - Prob. 2.45PCh. 2 - Prob. 2.46PCh. 2 - Prob. 2.47PCh. 2 - Prob. 2.49PCh. 2 - Prob. 2.50PCh. 2 - Prob. 2.51PCh. 2 - Prob. 2.52PCh. 2 - Prob. 2.53PCh. 2 - Prob. 2.54PCh. 2 - Prob. 2.58PCh. 2 - Prob. 2.63PCh. 2 - Prob. 2.64P
Knowledge Booster
Similar questions
- The necessary math for this problem is provided on the last page. A quantum harmonic oscillator is in superposition of the ground state and the first excited state. The superposition is such, that when energy is measured, there's a 70% chance to obtain the ground state energy. a) Write down the wavefunction of this state, and express it in terms of m, ħ, and wo. See the last page before writing anything down. Briefly explain how you got all the factors and coefficients. b) An ensemble of such systems is prepared in this state. Measurements of energy are performed. What is the expectation value of E3 (that's energy cubed)? There is an easy way of doing this that does not involve any integrals. Whichever method you pursue, make sure you explain your steps. This should not take a lot of space (use notation such as wn, En, etc. - no need to write them out explicitly). We mostly care about a clear explanation of your set-up.arrow_forwardH. W Solve the time-independent Schrödinger equation for an infinite square well with a delta-function barrier at the center: | a8(x). for (-aarrow_forward2.1 Illustrate with labels the eigenvalues of a harmonic oscillator potential. 2.2 The expectation value of the energy of the harmonic oscillator potential satisfy the inequality (AE) 2 am(Ax) +mw?(Ax)?. Show that the minimum energy of this potential satisfies (E) > hw. 2.3 A diatomic carbon monoxide molecule has a reduced mass of u = 6.85 u. It experiences an infrared transition a = 4.6 µm. Calculate the strength of its bond.arrow_forwardProblem 3.10 Is the ground state of the infinite square well an eigenfunction of momentum? If so, what is its momentum? If not, why not?arrow_forwardProblem 4. 1. Find the energy and the wave function for a particle moving in an infinite 1 = 0. spherical well of radius a with I Hint: Replace (r) → f(r)/r.arrow_forwardConsider the half oscillator" in which a particle of mass m is restricted to the region x > 0 by the potential energy U(x) = 00 for a O where k is the spring constant. What are the energies of the ground state and fırst excited state? Explain your reasoning. Give the energies in terms of the oscillator frequency wo = Vk/m. Formulas.pdf (Click here-->)arrow_forward4.8. The energy eigenfunctions V1, V2, V3, and 4 corresponding to the four lowest energy states for a particle confined in the finite potential well = {0 V(x) = -Vo |x| a/2 are sketched in Fig. 4.4. For which of these energy eigenfunctions would the probability of finding the particle outside the well, that is, in the region [x] > a/2, be greatest? Explain. Justify your reasoning using the solution to the Schrödinger equation in the region x > a/2.arrow_forwardInfinite/finite Potential Well 1. Sketch the solution (Wave function - Y) for the infinite potential well and show the following: (a) Specify the boundary conditions for region I, region II, and region III. (i.e. U = ?, and x = ?) (b) Specify the length of the potential well (L=10 cm) (c) Which region will have the highest probability of finding the particle?arrow_forwardProblem 1.17 A particle is represented (at time=0) by the wave function A(a²-x²). if-a ≤ x ≤+a. 0, otherwise. 4(x, 0) = { (a) Determine the normalization constant A. (b) What is the expectation value of x (at time t = 0)? (c) What is the expectation value of p (at time t = 0)? (Note that you cannot get it from p = md(x)/dt. Why not?) (d) Find the expectation value of x². (e) Find the expectation value of p².arrow_forwardA particle of mass in moving in one dimension is confined to the region 0 < 1 < L by an infinite square well potential. In addition, the particle experiences a delta function potential of strengtlh A located at the center of the well (Fig. 1.11). The Schrödinger equation which describes this system is, within the well, + A8 (x – L/2) v (x) == Ep(x), 0 < x < L. !! 2m VIx) L/2 Fig. 1.11 Find a transcendental equation for the energy eigenvalues E in terms of the mass m, the potential strength A, and the size L of the system.arrow_forwardA neutron of mass m with energy E a,V(x) =+Vo I. Draw the potential sketch! Write down the Schrödinger equation for: II. region I (0 a,V(x) =+Vo ), Calculate the total probability of nucleons tunneling through the barrier?; from region I to II (with E 2 0 and E < 0). II.arrow_forward3.1 Illustrate with labels the infinite square-well potential. 3.2 Mention 4 noteworthy properties of an infinite potential well.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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