Pearson eText -- Physics for Scientists and Engineers with Modern Physics -- Instant Access (Pearson+)
5th Edition
ISBN: 9780137488179
Author: Douglas Giancoli
Publisher: PEARSON+
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Chapter 38, Problem 23P
To determine
The proof that for a particle in a perfectly rigid box, the wavelength of the wave function for any state is the de Broglie wavelength.
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(10) i) Use the quantum mechanical kinetic energy operator T, =
to find the kinetic energy of the first
2m dx?
excited state of the Particle in a Box (with n = 2). ii) Then, use the relationship of kinetic energy and momentum (KE
= p?/2m) to find an equation for the de Broglie wavelength of the particle in a one-dimensional box as a function only
of the box length L and quantum number n. iii) Sketch the wavefunction in the box to verify that the expression you
obtained in part ii) is correct.
(i) We consider a one-dimensional potential barrier problem. In order for the particle to tunnel
through the potential barrier of the width L, the difference between the barrier height U and the
incident energy E of the particle with mass m has to be close. Using the transmission
probability given in the text book / lecture, obtain the energy difference U-E which gives the
transmission probability of exp(-2).
(ii) We consider an infinite square well potential with the width L. Obtain the energy E_{gr} of
the lowest energy level (ground state) of the particle with mass m, and show that E_{gr} scales
linearly with E-U in the problem (i). The potential structures of (i) and (ii) can be viewed as
"shadows" of each other.
Energy
U
---E«
Electron
X
L
L
(iii) We now consider a 3-dimensional infinite square well potential having the length of the x, y, and z
directions to be all L. V=L**3 is the volume of the cube of this potential. We consider energy level of a
single particle (boson)…
(i) We consider a one-dimensional potential barrier problem. In order for the particle to tunnel
through the potential barrier of the width L, the difference between the barrier height U and the
incident energy E of the particle with mass m has to be close. Using the transmission
probability given in the text book / lecture, obtain the energy difference U-E which gives the
transmission probability of exp(-2).
(ii) We consider an infinite square well potential with the width L. Obtain the energy E_{gr} of
the lowest energy level (ground state) of the particle with mass m, and show that E_{gr} scales
linearly with E-U in the problem (i). The potential structures of (i) and (ii) can be viewed as
"shadows" of each other.
Energy
U
---E-
Electron
X
L
L
Chapter 38 Solutions
Pearson eText -- Physics for Scientists and Engineers with Modern Physics -- Instant Access (Pearson+)
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