Modern Physics
Modern Physics
2nd Edition
ISBN: 9780805303087
Author: Randy Harris
Publisher: Addison Wesley
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Chapter 6, Problem 20E

(a)

To determine

The change in the position and the kinetic energy of a particle, classically.

(b)

To determine

The complete wave function everywhere.

(c)

To determine

The probability of the reflection of the particles.

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This question is for modern physics and wave and particle:   (a) To how small a region must an electron be confined for borderline relativistic speeds – say, 0.05c – to become reasonably likely? (Ans: 3.9×10^−12m ) (b) On the basis of this, would you expect relativistic effects to be prominent for hydrogen’s electron, which has an orbit radius near 10-10? For a lead atom “inner-shell” electron of orbit radius 10-12m?
For ultrarelativistic particles such as photons or high-energy electrons, the relation between energy and momentum is not E = p2/2m but rather E = pc. (This formula is valid for massless particles, and also for massive particles in the limit E » mc2.) Estimate the minimum energy of an electron confined inside a box of width 10-15 m. It was once thought that atomic nuclei might contain electrons; explain why this would be very unlikely.
Problem 1: (a) A non-relativistic, free particle of mass m is bouncing back and forth between two perfectly reflecting walls separated by a distance L. Imagine that the two oppositely directed matter waves associated with this particle interfere to create a standing wave with a node at each of the walls. Find the kinetic energies of the ground state (first harmonic, n = 1) and first excited state (second harmonic, n = 2). Find the formula for the kinetic energy of the n-th harmonic. (b) If an electron and a proton have the same non-relativistic kinetic energy, which particle has the larger de Broglie wavelength? (c) Find the de Broglie wavelength of an electron that is accelerated from rest through a small potential difference V. (d) If a free electron has a de Broglie wavelength equal to the diameter of Bohr's model of the hydrogen atom (twice the Bohr radius), how does its kinetic energy compare to the ground-state energy of an electron bound to a Bohr model hydrogen atom?
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