Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
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Chapter 8.1, Problem 8.2P
To determine
The best bound on for the one-dimensional harmonic oscillator using a trial wave function of the form
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Find the Probability (in %) that a particle trapped in a box, L units wide can be foundbetween 0.25L and 0.5L for the ground state.Normalize the wave function given by ψ (x) = A exp(−ax2) over the domain −∞ ≤x ≤ +∞ where A and a are constants.
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Introduction To Quantum Mechanics
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- Find the expectation value of the kinetic energy for the particle in the state, (x,t)=Aei(kxt). What conclusion can you draw from your solution?arrow_forwardFind the expectation value x2 of the square of the position for a quantum harmonic oscillator in the ground state. Note: +dxx2ea x 2=(2a 3/2)1.arrow_forwardFind the expectation value of the square of the momentum squared for the particle in the state, (x,t)=Aei(kxt). What conclusion can you draw from your solution?arrow_forward
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- Solve the 3-dimensional harmonic oscillator for which V(r) = 1/2 mω2(x2 + y2 + z2), by the separation of variables in Cartesian coordinates. Assume that the 1-D oscillator has eigenfunctions ψn(x) that have corresponding energy eigenvalues En = (n+1/2)ħω. What is the degeneracy of the 1st excited state of the oscillator?arrow_forwardA free electron has a kinetic energy 13.3eV and is incident on a potential energy barrier of U =32.1eV and width w =0.091nm. What is the probability for the electron to penetrate this barrier (in %)? Check the correct answer and show all workarrow_forwardConsider a particle of mass m, located in a potential energy well.one-dimensional (box) with infinite height walls. The wave function that describes this system is:Ψn(x) = K sin (nπx /L), for 0 ≤ x ≤ LΨn(x) = 0 for any other value.K is a constant and n = 1,2,3,... Determine K*K = │K│2arrow_forward
- Check the uncertainty principle for the wave function. *please pay attention to the picturearrow_forwardA one-dimensional harmonic oscillator wave function is ψ = Axe-bx2(a) Show that ψ satisfies as shown. (b) Find b and the total energy E. (c) Is this wave function for the ground state or for the first excited state?arrow_forward
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