# Lect13 - Lecture 13 Superposition Time-Dependent Quantum...

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Lecture 13, p 1 Time-Dependent Quantum States x | ψ (x,t 0 )| 2 U= U= 0 x L | ψ (x,t=0)| 2 U= U= 0 x L

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Lecture 13, p 2 Last Week Time-independent 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): Finite-depth square well Boundary conditions. Particle can “leak” into forbidden region. Comparison with infinite-depth 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 = + -
Lecture 13, p 3 Today Time dependent SEQ: Superposition of states and particle motion Measurement in quantum physics Time-energy uncertainty principle

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Lecture 13, p 4
Lecture 13, p 5 Time-Dependent 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 time-dependent 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 time-dependent 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 =

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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. a. Draw an Argand diagram of e i θ . The Argand diagram of a complex number, A, puts Re(A) on the x-axis and Im(A) on the y-axis. Notice the trig relation between the x and y components. θ is the angle of A from the real axis. In an Argand diagram, e i θ looks like a vector of length 1, and components (cos θ , sin θ ). Re(A) Im(A) θ A = e i θ The quantity, ce i θ (c and θ both real), is a complex number of magnitude |c|. The magnitude of a complex number, A, is |A| = (A*A), where A* is the complex conjugate of A. Re(A) Im(A) θ = ϖ t A = e i ϖ t b. Suppose that θ varies with time, θ = ϖ t. How does the Argand diagram behave? At t = 0, θ = 0, so A = 1 (no imaginary component). As time progresses, A rotates counterclockwise with angular frequency ϖ . This is the math that underlies phasors.
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Lect13 - Lecture 13 Superposition Time-Dependent Quantum...

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