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Physics 451 Quantum mechanics I Fall 2012 Sep 17, 2012 Karine Chesnel Quantum mechanics Announcements • Homework this week: Tuesday Sep 18 by 7pm: HW # 6 pb 2.10, 2.11, 2.12, 2.13, 2.14 Thursday Sep 20 by 7pm: HW # 7 pb 2.19, 2.20, 2.21, 2.22 • Monday: Review- Practice test Plan to work on your selected problem with your group and prepare the solution to be presented in class (~ 5 to 7 min) Quantum mechanics No student assigned to the following transmitters 17A79020 1E71A9C6 Please register your i-clicker at the class website! Quantum mechanics Ch 2.3 Harmonic oscillator V(x) Solving the Schrödinger equation: 1  2 2 p   m x    E  2m  Expressing the Hamiltonian a  in terms of convenient operators: 1  H    a a    2  or x 1 2m   ip  m x  1  H    a a    2  Quantum mechanics Ch 2.3 Harmonic oscillator Ladder operators: Raising operator: a n  n  1 n1 Lowering operator: a n  n n1 In this definition, the states  n are normalized a  n 1 a n  n1 En   En En   Quantum mechanics Quiz 8a What will be the final state of the particle after applying this operator? (ignore the coefficients) a a  a   a   n A.  n B.  n1 C.  n 2 D.  n1 E.  n 3 2 3 Quantum mechanics Ch 2.3 Harmonic oscillator Stationary states The ground state is given by the condition a 0  1  ip  m x  0  0 2m  m  m   2  0  x    e   1/4 Ground energy a 0  0 E0  1  2 x2 1 n n   a   0 n! 1  En   n    2  Quantum mechanics Ch 2.3 Harmonic oscillator Stationary states m  m   2  0  x    e   1/4 n  1 n  a   0 n! x2 The stationary states are orthonormal *   m n   nm Hermite polynomials Pb 2.10 Building  0  1 and  2 Checking the orthogonality Quantum mechanics Ch 2.3 Harmonic oscillator Energy levels V(x) 1  En   n    2  x Quantum mechanics Ch 2.3 Harmonic oscillator Expressing x, p and H in terms of ladder operators: a  1 2m   ip  m x  • Operator position x • Operator momentum pi • Operator Hamiltonian 2m m 2  a  a   a  a  1 1   H    a a      a a   2 2   Quantum mechanics Ch 2.3 Harmonic oscillator V(x) Expectation values x  a  2m   a  p i m 2 a  a   V  1  m 2 x 2  2 4 T  1 2  p  2m 4 ^   a  a   a  a  H   cn En n 1 2 2 2  x   a    a  2 2m  a    a  2 2m 2 2  a a  a a  a a  a a Pb 2.11, 2.12 Quantum mechanics Quiz 8b a  1 2m   ip  m x  Since the operators a+ and a- are shifting the stationary states from one level to another, and since the stationary states are all orthogonal, the expectations values for x and p on any state will ALWAYS be zero! A. True B. False Pb 2.13

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Download Sep 17 - BYU Physics and Astronomy