Physics for Scientists and Engineers
6th Edition
ISBN: 9781429281843
Author: Tipler
Publisher: MAC HIGHER
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Question
Chapter 35, Problem 36P
(a)
To determine
The solution of the Schrödinger equation is
(b)
To determine
The wave function
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A function of the form e^−gx2 is a solution of the Schrodinger equation for the harmonic oscillator, provided that g is chosen correctly. In this problem you will find the correct form of g.
(a) Start by substituting Ψ = e^−gx2 into the left-hand side of the Schrodinger equation for the harmonic oscillator and evaluating the second derivative.
(b) You will find that in general the resulting expression is not of the form constant × Ψ, implying that Ψ is not a solution to the equation. However, by choosing the value of g such that the terms in x^2 cancel one another, a solution is obtained. Find the required form of g and hence the corresponding energy.
(c) Confirm that the function so obtained is indeed the ground state of the harmonic oscillator and has the correct energy.
The following Eigen function is a typical solution of the time-independent Schrödinger equation and
satisfies boundary conditions for a particle in a confined space of a certain length.
y(x) = sin
(~77)
(a) Plot the wave function as a function of x for L = 30 cm and n = 1, 2, 3 and 4.
Note: You will need to have 4 plots in the same graph.
(b) On a separate graph, plot the probability density (112) as a function of x using the conditions
specified in part (a). Note: You will need to have 4 plots in the same graph.
(c) Report your observations for parts (a) and (b)
A particle of mass m is trapped in a three-dimensional rectangular potential well with sides of length L, L/ √2, and 2L. Inside the box V = 0, outside V = ∞. Assume that Ψ = Asin (k1x) sin (k2y) sin (k3z) inside the well. Substitute this wave function into the Schrödinger equation and apply appropriate boundary conditions to find the allowed energy levels. Find the energy of the ground state and first four excited levels. Which of these levels are degenerate?
Chapter 35 Solutions
Physics for Scientists and Engineers
Ch. 35 - Prob. 1PCh. 35 - Prob. 2PCh. 35 - Prob. 3PCh. 35 - Prob. 4PCh. 35 - Prob. 5PCh. 35 - Prob. 6PCh. 35 - Prob. 7PCh. 35 - Prob. 8PCh. 35 - Prob. 9PCh. 35 - Prob. 10P
Ch. 35 - Prob. 11PCh. 35 - Prob. 12PCh. 35 - Prob. 13PCh. 35 - Prob. 14PCh. 35 - Prob. 15PCh. 35 - Prob. 16PCh. 35 - Prob. 17PCh. 35 - Prob. 18PCh. 35 - Prob. 19PCh. 35 - Prob. 20PCh. 35 - Prob. 21PCh. 35 - Prob. 22PCh. 35 - Prob. 23PCh. 35 - Prob. 24PCh. 35 - Prob. 25PCh. 35 - Prob. 26PCh. 35 - Prob. 27PCh. 35 - Prob. 28PCh. 35 - Prob. 29PCh. 35 - Prob. 30PCh. 35 - Prob. 31PCh. 35 - Prob. 32PCh. 35 - Prob. 33PCh. 35 - Prob. 34PCh. 35 - Prob. 35PCh. 35 - Prob. 36PCh. 35 - Prob. 37PCh. 35 - Prob. 38P
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