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# hw8 - ∣ n,l,m 〉 = ∣ n,n − 1 m 〉 for the hydrogen...

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Assignment #8 PHYS-446 due in class on 28 November 2008 1. (Griffiths, Problem 3.15) Show that two non-commuting operators cannot have a complete set of common eigenfunctions. Hint: Show that if P and Q have a complete set of common eigenfunctions, then [ P , Q ] f = 0 for any function in Hilbert space. 2. (Griffiths, Problem 4.1) (a) Work out all of the canonical commutation relations for components of the operators r and p : [ x , y ] , [ x , p y ] , [ x , p x ] , [ p y , p z ] and so on. 3. Consider the states of maximum angular momentum
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Unformatted text preview: ∣ n,l,m 〉 = ∣ n,n − 1, m 〉 for the hydrogen atom. (a) Show that 〈 n,n − 1, m ∣ 1 r 2 ∣ n,n − 1, m 〉 = 2 2n − 1 n 3 a 2 (b) Calculate 〈 n,n − 1, m ∣ r 2 ∣ n,n − 1, m 〉 4. (Griffiths, Problem 4.14) 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....
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