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Quantum Mechanics: Stationary States and Wave Functions

Explore time-independent Schrödinger equation, stationary states properties, and quantum mechanics principles in Ch. 2.1 and Ch. 2.2, including solutions in infinite square wells and harmonic oscillators. Learn about energy quantization and probability measurements in quantum mechanics.

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Quantum Mechanics: Stationary States and Wave Functions

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  1. Physics 451 Quantum mechanics I Fall 2012 Sep 12, 2012 Karine Chesnel

  2. Quantum mechanics Announcements • Homework remaining this week: • Extended Friday Sep 14 by 7pm: • HW # 5 Pb 2.4, 2.5, 2.7, 2.8 Note: Penalty on late homework: - 2pts per day Credit for group presentations: Homework 2: 20 points Quiz 5: 5 points

  3. Quantum mechanics Announcements No student assigned to the following transmitters: 2214B68 17A79020 1E5C6E2C 1E71A9C6 Please register your i-clicker at the class website!

  4. Quantum mechanics Ch 2.1 Time-independent Schrödinger equation • Space dependent part: Solution y(x) depends on the potential function V(x). Stationary state Associated to energy E

  5. Quantum mechanics Ch 2.1 with is independent of time The expectation value for the momentum is always zero In a stationary state! are zero!) (Side note: does not mean that and Stationary states Properties: • Expectation values are not changing in time (“stationary”):

  6. Quantum mechanics Ch 2.1 Stationary states Properties: • Hamiltonian operator - energy

  7. Quantum mechanics Ch 2.1 where • Associated expectation value for energy Stationary states • General solution

  8. A particle, is in a combination of stationary states: What will we get if we measure its energy? Quantum mechanics Quiz 6a • one of the values

  9. A particle, is in a combination of stationary states: What is the probability of measuring the energy En? Quantum mechanics Quiz 6b • 0

  10. Quantum mechanics Ch 2.2 Time-independent potential Expectation value for the energy:

  11. Quantum mechanics Ch 2.2 The particle can only exist in this region Infinite square well V(x)=0 for 0<x<a V=∞ else 0 a x Shape of the wave function?

  12. Quantum mechanics Ch 2.2 with Infinite square well Solutions to Schrödinger equation: Simple harmonic oscillator differential equation

  13. Quantum mechanics Ch 2.2 At x=0: At x=a: with Infinite square well Solutions to Schrödinger equation: Boundary conditions:

  14. Quantum mechanics Ch 2.2 Infinite square well Possible states and energy values: Quantization of the energy Each state yn is associated to an energy En

  15. Quantum mechanics Ch 2.2 Excited states Ground state 3. They are orthonormal Infinite square well Properties of the wave functions yn: 1.They are alternatively even and odd around the center 2. Each successive state has one more node 0 a x 4. Each state evolves in time with the factor

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

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

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