Chapter 7: Q12CQ (page 278)
Question: A particle is trapped in a spherical infinite well. The potential energy is 0for r < a and infinite for r > a . Which, if any, quantization conditions would you expect it to share with hydrogen, and why?
Answer:
The particle will have the same quantization which is found in hydrogen.
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Exercise 80 discusses the idea of reduced mass. When two objects move under the influence of their mutual force alone, we can treat the relative motion as a one particle system of mass . Among other things, this allows us to account for the fact that the nucleus in a hydrogen like atom isnโt perfectly stationary, but in fact also orbits the centre of mass. Suppose that due to Coulomb attraction, an object of mass and charge orbits an object of mass and charge +Ze . By appropriate substitution into formulas given in the chapter, show that (a) the allowed energies are , where is the hydrogen ground state, and (b) the โBohr Radiusโ for this system is ,where is the hydrogen Bohr radius.
Question: Verify the correctness of the normalization constant of the radial wave function given in Table 7.4 as
Consider two particles that experience a mutual force but no external forces. The classical equation of motion for particle 1 is , and for particle 2 is , where the dot means a time derivative. Show that these are equivalent to , and .Where, .
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 โreduced massโ .
Taking thestates as representative, explain the relationship between the complexity numbers of nodes and antinodes-of hydrogen's standing waves in the radial direction and their complexity in the angular direction at a given value of n. Is it a direct or inverse relationship, and why?
At heart, momentum conservation is related to the universe being "translationally invariant," meaning that it is the same if you shift your coordinates to the right or left. Angular momentum relates to rotational invariance. Use these ideas to explain at least some of the differences between the physical properties quantized in the cubic three-dimensional box versus the hydrogen atom.
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