Quantum mechanics I Fall 2012 Physics 451 Quantum mechanics I Fall 2012 Sep 14, 2012 Karine Chesnel

Homework Friday Sep 14 by 7pm: HW # 5 pb 2.4, 2.5, 2.7, 2.8 Quantum mechanics Homework Friday Sep 14 by 7pm: HW # 5 pb 2.4, 2.5, 2.7, 2.8 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

No student assigned to the following transmitters Quantum mechanics No student assigned to the following transmitters 1E2B2F1A 1E5C6E2C 1E71A9C6 Please register your i-clicker at the class website!

Infinite square well Quantum mechanics Ch 2.2 Properties of the wave functions yn: 1.They are alternatively even and odd around the center Excited states 2. Each successive state has one more node 3. They are orthonormal Ground state a x 4. Each state evolves in time with the factor

Infinite square well Pb 2.4 Pb 2.5 Quantum mechanics Ch 2.2 Particle in one stationary state Pb 2.5 Particle in a combination of two stationary states evolution in time? oscillates in time expressed in terms of E1 and E2

Quiz 7a Could this function be the wave function Quantum mechanics Quiz 7a Could this function be the wave function of a particle in an infinite square well at a given time? x a Yes No Pb. 2.7

Decomposition of any wave function Quantum mechanics Ch 2.2 Infinite square well Decomposition of any wave function At time t = 0 Fourier’s series expansion

How to find the coefficients cn? Quantum mechanics Ch 2.2 Infinite square well How to find the coefficients cn? Dirichlet’s theorem Pb. 2.7 & 2.8

Expectation value for the energy: Quantum mechanics Ch 2.2 Infinite square well Expectation value for the energy: The probability that a measurement yields to the value En is Normalization

Quiz 7b Could this function be a solution for the wave function Quantum mechanics Quiz 7b Could this function be a solution for the wave function of a particle in an infinite square well at a given time? x a Yes No

Quantum mechanics Ch 2.3 Harmonic oscillator x V(x)

Solving the Schrödinger equation: Quantum mechanics Ch 2.3 Harmonic oscillator Solving the Schrödinger equation: x V(x)

Harmonic oscillator Quantum mechanics Ch 2.3 x V(x) Expressing the Hamiltonian in terms of convenient operators: Commutator: or

Harmonic oscillator Quantum mechanics Ch 2.3 Ladder operators: If y is a solution the Schrödinger equation for energy E Then is a solution the Schrödinger equation for energy Quantization of energy And is a solution the Schrödinger equation for energy We can built all the solutions just starting from one solution (ground level)

Starting from ground state Quantum mechanics Ch 2.3 Harmonic oscillator Ladder operators: Raising operator: Lowering operator: Starting from ground state