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Modern Physics
Found in: Page 278
Modern Physics

Modern Physics

Book edition 2nd Edition
Author(s) Randy Harris
Pages 633 pages
ISBN 9780805303087

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Short Answer

Question: Explain to your friend. who has just learned about simple one-dimensional standing waves on a string fixed at its ends, why hydrogen's electron has only certain energies, and why, for some of those energies, the electron can still be in different states?

Answer:

If the energy levels are limited, still waves can have different orientations and may result in a different state but of the same energy.

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Significance of the one-dimensional standing waves

One-dimensional standing waves can be observed when a medium is having its opposite ends fixed, and nodes are located at the endpoints. The simplest example of the one-dimensional standing wave will be the one having only one antinode in the middle. This will be half of the wavelength.

Step 3: Identification of reason for electrons having the same energy but different states

The electron of hydrogen behaves as a wave of the bound state, that is why they have only certain energy or certain allowed standing waves. But when multiple dimensions are introduced in space, it is possible to have different standing waves but still have the same frequency/energy.

Assume a square wave of two dimensions for instance, the energy/frequency of the wave will be the same for both the following conditions,

(i) when the wave contains two waves along –axis and one wave along -axis

(ii) when the wave contains two waves along -axis and one wave along –axis.

Thus, if the energy levels are limited, still waves can have different orientations and may result in a different state but of the same energy.

Most popular questions for Physics Textbooks

Consider two particles that experience a mutual force but no external forces. The classical equation of motion for particle 1 is v˙1=F2on1/m1, and for particle 2 is v˙2=F1on2/m2, where the dot means a time derivative. Show that these are equivalent to v˙cm=constant, and v˙rel=FMutual/μ . Where, v˙cm=(m1v˙1+m2v˙2)/(m1m2),FMutual=-Fion2 and μ=m1m2(m1+m2).

In other words, the motion can be analyzed into two pieces the center of mass motion, at constant velocity and the relative motion, but in terms of a one-particle equation where that particle experiences the mutual force and has the “reduced mass” μ.

For a hydrogen atom in the ground state. determine (a) the most probable location at which to find the electron and (b) the most probable radius at which to find the electron, (c) Comment on the relationship between your answers in parts (a) and (b).

Classically, it was expected that an orbiting electron would emit radiation of the same frequency as its orbit frequency. We have often noted that classical behaviour is observed in the limit of large quantum numbers. Does it work in this case? (a) Show that the photon energy for the smallest possible energy jump at the “low-n-end” of the hydrogen energies is 3|E0|/n3, while that for the smallest jump at the “high-n-end” is 2|E0|/n3, where E0 is hydrogen’s ground-state energy. (b) Use F=ma to show that the angular velocity of a classical point charge held in orbit about a fixed-point charge by the coulomb force is given by ω=e2/4πε0mr3. (c) Given that r=n2a0, is this angular frequency equal to the minimum jump photon frequency at either end of hydrogen’s allowed energies?

Classically, an orbiting charged particle radiates electromagnetic energy, and for an electron in atomic dimensions, it would lead to collapse in considerably less than the wink of an eye.

(a) By equating the centripetal and Coulomb forces, show that for a classical charge -e of mass m held in a circular orbit by its attraction to a fixed charge +e, the following relationship holds

ω=er-3/24πε0m .

(b) Electromagnetism tells us that a charge whose acceleration is a radiates power P=e2a2/6ε0c3. Show that this can also be expressed in terms of the orbit radius as P=e696π2ε03m2c3r4. Then calculate the energy lost per orbit in terms of r by multiplying the power by the period T=2π/ω and using the formula from part (a) to eliminate .

(c) In such a classical orbit, the total mechanical energy is half the potential energy, or Eorbit=-e28πε0r. Calculate the change in energy per change in r : dEorbit/dr. From this and the energy lost per obit from part (b), determine the change in per orbit and evaluate it for a typical orbit radius of 10-10 m. Would the electron's radius change much in a single orbit?

(d) Argue that dividing dEorbit/dr by P and multiplying by dr gives the time required to change r by dr . Then, sum these times for all radii from rinitial to a final radius of 0. Evaluate your result for rinitial=10-10 m. (One limitation of this estimate is that the electron would eventually be moving relativistically).

For an electron in the (n,l,ml)=(2,0,0) state in a hydrogen atom, (a) write the solution of the time-independent Schrodinger equation,

(b) verify explicitly that it is a solution with the expected angular momentum and energy.

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