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Unformatted text preview: Lecture 13, p 1 “But why must I treat the measuring device classically? What will happen to me if I d o n ’ t ??”Eugene Wigner “There is obviously no such limitation – I can measure the energy and look at my watch; then I know both energy and time!”L. D. Landau, on the timeenergy uncertainty principle “When I hear of Schrödinger’s cat, I reach for my gun.”Stephen W. Hawking Lecture 13, p 2 Lab 3 Comments Lab 3 meets this week if you are normally in 132 Loomis. Lab 3 meets next week if you are normally in 164 Loomis. So does Discussion, and there is a quiz , so don’t skip... For the lab: You will need your “Active Directory” Login See: http://www.ad.uiuc.edu You can save a lot of time by reading the lab ahead of time. It’s a tutorial on how to draw wave functions Lecture 13, p 3 Lecture 13: Superposition & TimeDependent Quantum States x  ψ (x,t ) 2 U= ∞ U= ∞ x L  ψ (x,t=0) 2 U= ∞ U= ∞ x L Lecture 13, p 4 Last Week Timeindependent Schrodinger’s Equation (SEQ): • It describes a particle that has a definite energy, E . • The solutions, ψ (x), are time independent ( stationary states ). We considered two potentials, U(x): Finitedepth square well • Boundary conditions. • Particle can “leak” into forbidden region. • Comparison with infinitedepth well. Harmonic oscillator • Energy levels are equally spaced. • A good approximation in many problems. ) ( ) ( ) ( ) ( 2 2 2 2 x E x x U dx x d m ψ ψ ψ = + ℏ Lecture 13, p 5 Today Time dependent SEQ: Superposition of states and particle motion Measurement in quantum physics Schrödinger’s cat Timeenergy uncertainty principle Lecture 13, p 6 TimeDependent SEQ To explore how particle wave functions evolve with time, which is useful for a number of applications as we shall see, we need to consider the timedependent SEQ : Changes from the time independent version: • E ψ → i ħ d Ψ /dt We no longer assume a definite E. • ψ (x) → Ψ (x,t) The solutions will have time dependence. • i = √ (1) appears The solutions will be complex. This equation describes the complete time and space dependence of a quantum particle in a potential U(x). It replaces the classical particle dynamics law, F=ma. The SEQ is linear in Ψ , and so the Superposition Principle applies : If Ψ 1 and Ψ 2 are solutions to the timedependent SEQ, then so is any linear combination of Ψ 1 and Ψ 2 (example: Ψ = 0.6 Ψ 1 + 0.8 i Ψ 2 ) 2 2 2 ( , ) ( , ) ( ) ( , ) 2 d x t d x t U x x t i m dx dt Ψ Ψ + Ψ = ℏ ℏ Lecture 13, p 7 Review of Complex Numbers The equation, e i θ = cos θ + isin θ , might be new to you. It is a convenient way to represent complex numbers. It also (once you are used to it) makes trigonometry simpler....
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 Spring '11
 MESTRE
 Magnetism, Energy, Particle, probability density, Superpositions

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