Modern Physics
3rd Edition
ISBN: 9781111794378
Author: Raymond A. Serway, Clement J. Moses, Curt A. Moyer
Publisher: Cengage Learning
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Question
Chapter 6, Problem 6P
To determine
The energy of the particle.
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PROBLEM 2. The potential energy of a weakly anharmonic oscillator can be
modeled by:
m
U(x)
P²+Bx*,
where the last quatric term describes a small anharmonic correction. The
energy levels En of the anharmonic oscillator in the first order in the pa-
rameter 3 are given by:
En = hw
5) + B(n|z*\n).
Calculate the energy of the ground state Eo of the anharmonic oscillator.
6.2. Solve the three-dimensional harmonic oscillator for
which
1
V(r) = -— mw² (x² + y²² +2²)
2
by separation of variables in Cartesian coordinates.
Assume that the one-dimensional oscillator has
eigenfunctions (x) with corresponding energy
eigenvalues En = (n + 1/2)hw. What is the degeneracy
of the first excited state of the oscillator?
The following problem arises in quantum mechanics (see Chapter 13, Problem 7.21). Find the number of ordered triples of nonnegative integers a, b, c whose sum a+b+c is a given positive integer n. (For example, if n = 2, we could have (a, b, c) = (2, 0, 0) or (0, 2, 0) or (0, 0, 2) or (0, 1, 1) or (1, 0, 1) or (1, 1, 0).) Hint: Show that this is the same as the number of distinguishable distributions of n identical balls in 3 boxes, and follow the method of the diagram in Example 5.
Chapter 6 Solutions
Modern Physics
Ch. 6.4 - Prob. 1ECh. 6.4 - Prob. 2ECh. 6.5 - Prob. 4ECh. 6.7 - Prob. 5ECh. 6.8 - Prob. 6ECh. 6 - Prob. 1QCh. 6 - Prob. 2QCh. 6 - Prob. 3QCh. 6 - Prob. 4QCh. 6 - Prob. 5Q
Ch. 6 - Prob. 6QCh. 6 - Prob. 7QCh. 6 - Prob. 8QCh. 6 - Prob. 1PCh. 6 - Prob. 2PCh. 6 - Prob. 3PCh. 6 - Prob. 5PCh. 6 - Prob. 6PCh. 6 - Prob. 7PCh. 6 - Prob. 8PCh. 6 - Prob. 9PCh. 6 - Prob. 10PCh. 6 - Prob. 11PCh. 6 - Prob. 12PCh. 6 - Prob. 13PCh. 6 - Prob. 14PCh. 6 - Prob. 15PCh. 6 - Prob. 16PCh. 6 - Prob. 17PCh. 6 - Prob. 18PCh. 6 - Prob. 19PCh. 6 - Prob. 21PCh. 6 - Prob. 24PCh. 6 - Prob. 25PCh. 6 - Prob. 26PCh. 6 - Prob. 28PCh. 6 - Prob. 29PCh. 6 - Prob. 30PCh. 6 - Prob. 31PCh. 6 - Prob. 32PCh. 6 - Prob. 33PCh. 6 - Prob. 34PCh. 6 - Prob. 35PCh. 6 - Prob. 37PCh. 6 - Prob. 38P
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- Given that at time t = 0 a particle’s wave function is given by ψ(x, 0) =Ax/a, if 0 ≤ x ≤ a,A(b − x)/(b − a), if a ≤ x ≤ b, with A0, Otherwise.a and b as constants, answer the following questions; a) Find the normalization constant A in terms of the constants a and b. b) Sketch ψ(x, 0) as a function of x. c) Where is the particle most likely to be found at time t = 0? d) What is the probability of finding the particle to the left of a?arrow_forward2.4. 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)arrow_forward5.1 Consider a one-dimensional bound particle. Show that if the particle is in a stationary state at a given time, then it will always remain in a stationary state. 5.2 Consider a one-dimensional bound particlearrow_forward
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