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228s13-l17 - Physics 228 Today Ch 41 1-3 3D quantum...

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Physics 228 Today: Ch 41: 1-3: 3D quantum mechanics, hydrogen atom Website: Sakai 01:750:228 or www.physics.rutgers.edu/ugrad/228 Happy April Fools Day Monday, April 1, 2013
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Example / Worked Problems What is the ratio of the energy of the 3rd to the 2nd excited states of a harmonic oscillator? The harmonic oscillator energy levels are E n = (n+ ½ ) ħω , with ω 2 = k'/m. The ground state is n = 0, the second excited state is n = 2, and the third excited state is n = 3. Thus: r = E 3 /E 2 = (3+ ½ )/(2+ ½ ) = 7/5. What energy photon is given off when an electron in an infinite 1-D square well transitions from the 2nd excited state to the ground state? The infinite square well energy levels are E n = n 2 ħ 2 π 2 /2mL 2 . The ground state is n = 1, and the second excited state is n = 3. Thus the photon or transition energy is E γ = E n=3 - E n=1 = (3 2 -1 2 ) ħ 2 π 2 /2mL 2 = 8 ħ 2 π 2 /2mL 2 = 4 ħ 2 π 2 /mL 2 . Monday, April 1, 2013
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Example / Worked Problems What is the ratio of the energy of the 3rd to the 2nd excited states of a harmonic oscillator? The ground state is n = 0. What energy photon is given off when an electron in an infinite 1-D square well transitions from the 2nd excited state to the ground state? The ground state is n = 1. While we use n to denote the levels, note that for some potentials we use n = 0 for the ground state, while for other potentials we use n = 1 for the ground state. On a test, we will expect you to know n of the ground state. Monday, April 1, 2013
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Example / Worked Problems II What is the momentum of an electron in the 1 st excited state of an infinite square well potential? The infinite square well energy levels are E n = n 2 ħ 2 π 2 /2mL 2 . The first excited state is n = 2. The only way we know how to get the momentum is from the energy. The kinetic energy in this state is E 2 = 4 ħ 2 π 2 /2mL 2 = 2 ħ 2 π 2 / mL 2 . So we expect the magnitude of the momentum is p = (2mE) ½ = 2 π ħ /L = h/L. But is this the answer to the question? You need to be careful about how it is asked. If the "expectation value" of the momentum is requested, this is the signed momentum, not the magnitude, and the answer is 0. Since the particle in a box isn't going anywhere, its average momentum must be 0, although the average magnitude of the momentum is non-0. Be careful about whether the average momentum or the magnitude of the momentum is requested.
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