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FinalWi08Solutions

FinalWi08Solutions - Final Examination Physics 115B Winter...

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Final Examination March 20, 2008 Physics 115B, Winter 2008 E. Abers Final Examination Solutions and Comments Some possibly useful formulas are listed at the end. Question 1 [20 Points] Harmonic Oscillator A particle moves in a one-dimensional harmonic oscillator potential: H = p 2 2 m + 2 x 2 2 and is known to be in the state | ψ ) = 1 2 bracketleftBig | ψ o ) - i | ψ 1 ) bracketrightBig where | ψ n ) is the eigenstate of the n ’th energy level. What is the expectation value ( ψ | p | ψ ) of the momentum p in this state? GO ON TO THE NEXT PAGE SOLUTION From table 5 at the end of the exam, a - - a + = 2 ip 2 m planckover2pi1 ω so p = i radicalbigg m planckover2pi1 ω 2 ( a + - a - ) and p | ψ ) = i radicalbigg m planckover2pi1 ω 2 ( a + - a - ) | ψ o ) - i | ψ 1 ) 2 = i 2 planckover2pi1 ωm bracketleftBig | ψ 1 ) - 2 i | ψ 2 ) - 0 + i | ψ 0 ) bracketrightBig and ( ψ | p | ψ ) = i 2 planckover2pi1 ωm bracketleftBig ( ψ | ψ 1 ) - i 2 ( ψ | ψ 2 ) + i ( ψ | ψ 0 ) bracketrightBig = i 2 planckover2pi1 ωm bracketleftBig ( ψ 1 | ψ ) * - i 2 ( ψ 2 | ψ ) * + i ( ψ 0 | ψ ) * bracketrightBig
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2 Final Examination Now ( ψ 1 | ψ ) = - i 2 ( ψ 0 | ψ ) = 1 2 ( ψ 2 | ψ ) = 0 so ( ψ | p | ψ ) = i 2 radicalbigg planckover2pi1 ωm 2 (+ i + i ) = - radicalbigg planckover2pi1 ωm 2 The answer is real (because p is Hermitean), and it is not zero. Many of you didn’t take the complex conjugate correctly in computing the scalar product. An almost identical question appeared on the midterm.
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Physics 115B, Winter 2008 3 Question 2 [10 Points] Tritium and Helium-3 The nucleus of a tritium atom is made up of one proton and two neutrons. Tritium is unstable. The nucleus decays into a 3 He, made up of two protons and one neutron. Suppose an electron is in the ground state of a tritium atom. Just after the nucleus decays, the energy of the electron is measured. What is the lowest possible result of this measurement? What is the probability that the electron is has this energy? Assume the nuclear charge changes so suddenly that the electron wave function is unchanged when the measurement is made. Also, treat the nuclei as infinitely heavy compared to the electron – don’t worry about center of mass corrections. GO ON TO THE NEXT PAGE SOLUTION The electron wave function is the same as for hydrogen: R ( r ) = 2 a 3 / 2 e - r/a For Helium, Z = 2. The ground state energy is E 1 = - 4 α 2 mc 2 2 In the radial wave function replace α by 2 α , or equivalently a by a/ 2: and the He radial wave function is R 10 = 2 parenleftbigg 2 a parenrightbigg 3 / 2 e - 2 r/a The probability we are looking for is P = vextendsingle vextendsingle vextendsingle vextendsingle integraldisplay 0 R ( r ) R 10 ( r ) r 2 dr vextendsingle vextendsingle vextendsingle vextendsingle 2 = 128 a 6 vextendsingle vextendsingle vextendsingle vextendsingle integraldisplay 0 e - r/a e - 2 r/a r 2 dr vextendsingle vextendsingle vextendsingle vextendsingle 2 = 128 a 6 vextendsingle vextendsingle vextendsingle vextendsingle integraldisplay 0 e - 3 r/a r 2 dr vextendsingle vextendsingle vextendsingle vextendsingle 2 = 4 128 a 6 parenleftBig a 3 parenrightBig 6 = 2 9 3 6 0 . 702 ...
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