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 6.8, Problem 6.30P
(a)
To determine
Prove that the solution of Schrodinger equation can be expressed in the form
(b)
To determine
Obtain the function
(c)
To determine
Obtain
(d)
To determine
Obtain the function
(e)
To determine
Obtain the function
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Check out a sample textbook solutionStudents have asked these similar questions
Consider the Schrodinger equation for a one-dimensional linear harmonic oscillator:
-(hbar2/2m) * d2ψ/dx2 + (kx2/2)*ψ(x) = Eψ(x)
Substitute the wavefunction ψ(x) = e-(x^2)/(ξ^2) and find ξ and E required to satisfy the Schrodinger equation. [Hint: First calculate the second derivative of ψ(x), then substitute ψ(x) and ψ′′(x). After this substitution, there will be an overall factor of e-(x^2)/(ξ^2) on both sides of the equation which canbe an canceled out. Then, gather all terms which depend on x into one coefficient multiplying x2. This coefficient must be zero because the equation must be satisfied for any x, and equating it with zero yields the expression for ξ. Finally, the remaining x-independent part of the equation determines the eigenvalue for energy E associated with this solution.]
a) Write down the one-dimensional time-dependent Schro ̈dinger equation for a particle of
mass m described by a wavefunction Ψ(t, x) in a potential V (x). b) For energy eigenstates, the wavefunction can be written as
Ψ(t, x) = f (t) ψ(x).
For this wavefunction:(i) state the time-independent equation that must be satisfied by ψ(x).
(ii) derive an expression for f(t), in terms of the energy of the particle E.
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.
Chapter 6 Solutions
Introduction To Quantum Mechanics
Ch. 6.1 - Prob. 6.1PCh. 6.2 - Prob. 6.2PCh. 6.2 - Prob. 6.3PCh. 6.2 - Prob. 6.4PCh. 6.2 - Prob. 6.5PCh. 6.2 - Prob. 6.7PCh. 6.4 - Prob. 6.8PCh. 6.4 - Prob. 6.9PCh. 6.4 - Prob. 6.10PCh. 6.4 - Prob. 6.11P
Ch. 6.4 - Prob. 6.12PCh. 6.4 - Prob. 6.13PCh. 6.5 - Prob. 6.14PCh. 6.5 - Prob. 6.15PCh. 6.5 - Prob. 6.16PCh. 6.5 - Prob. 6.17PCh. 6.6 - Prob. 6.18PCh. 6.6 - Prob. 6.19PCh. 6.7 - Prob. 6.20PCh. 6.7 - Prob. 6.21PCh. 6.7 - Prob. 6.22PCh. 6.7 - Prob. 6.23PCh. 6.7 - Prob. 6.25PCh. 6.8 - Prob. 6.26PCh. 6.8 - Prob. 6.27PCh. 6.8 - Prob. 6.28PCh. 6.8 - Prob. 6.30PCh. 6 - Prob. 6.31PCh. 6 - Prob. 6.32PCh. 6 - Prob. 6.34PCh. 6 - Prob. 6.35PCh. 6 - Prob. 6.36PCh. 6 - Prob. 6.37P
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