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Chapter 4: Q14P (page 156)
What is the most probable value of r, in the ground state of hydrogen? (The answer is not zero!) Hint: First you must figure out the probability that the electron would be found between r and r + dr.
The most probable value of r is in the ground state of hydrogen.
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[Refer to. Problem 4.59for background.] Suppose and, where and Kare constants.
(a) Find the fields E and B.
(b) Find the allowed energies, for a particle of mass m and charge q , in these fields, Answer: Comment: If K=0this is the quantum analog to cyclotron motion; is the classical cyclotron frequency, and it's a free particle in the z direction. The allowed energies,, are called Landau Levels.
An electron is in the spin state
(a) Determine the constant by normalizing .
(b) If you measured on this electron, what values could you get, and what is the probability of each? What is the expectation value of ?
(c) If you measured on this electron, what values could you get, and what is the probability of each? What is the expectation value of ?
(d) If you measured on this electron, what values could you get, and what is the probability of each? What is the expectation value of?
The raising and lowering operators change the value of m by one unit:
(4.120).
Where are constant. Question: What is , if the Eigen functions are to be normalized? Hint: First show thatis the Hermitian conjugate of (Since are observables, you may assume they are Hermitian…but prove it if you like); then use Equation 4.112.
Use Equation 4.32 to construct and . (You can take from Table 4.2, but you'll have to work out from Equations 4.27 and 4.28.) Check that they satisfy the angular equation (Equation 4.18), for the appropriate values of l and m .
(a) Work out all of the canonical commutation relations for components of the operator r and p : and so on.
(b) Confirm Ehrenfest’s theorem for 3 dimensions
(Each of these, of course, stand for three equations- one for each component.)
(c) Formulate Heisenberg’s uncertainty principle in three dimensions Answer:
But there is no restriction on, say,
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