Lecture 4

# Lecture 4 - Particles of Light Lecture 7 Matter Waves...

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1 Particles of Light Particles of Light Lecture 7: Matter Waves Reading: Ch. 39 Main Idea: Matter waves are governed by Schroedinger's equation. We discuss how it is used in quantum mechanics, then we apply it to a very simple quantum system that captures many behaviors of bound electrons in atoms.

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2 Inspiration for the quantum Inspiration for the quantum • Single particle diffraction (photons, electrons, neutons, etc.) • De Broglie relation assigns every object with momentum a wavelength that characterizes its wave behavior. = h / p
3 Quantum theory is intrinsically probabilistic Quantum theory is intrinsically probabilistic • We don't know where a particle is before it is measured → unlike Newton's kinematics where we could predict the particle's location at any moment given the initial condition • Motion in quantum mechanics is governed by a probability wave that, most importantly, displays interference.

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4 What is a quantum mechanical wavefunction? What is a quantum mechanical wavefunction? A wavefunction is a “field” that provides us a method of computing the “probability amplitudes” → a wavefunction obeys a certain wave equation The square of the wavefunction gives us a probability distribution also sometimes called a probability density  x ,t P x ,t =∣ x ,t ∣ 2 dV “wavefunction” 0 x ∣ 1 x ' ,t ∣ 2 dx ' Probability of measuring the particle between 0 and x: ∣ 1,0,0 r ∣ 2
5 Schroedinger's equation: equation for the wavefunction Schroedinger's equation: equation for the wavefunction • Like the E-field, probability waves are governed by a wave equation. Time-dependent Schroedinger Equation: • The wavefunction is not the probability wave. The probability of measuring the particle at time t between x and x+dx is 2 E x 2 = 1 c 2 2 E t 2 i ∂ t = −ℏ 2 2m 2 x 2 V x  Wave equation for E-field from Maxwell's equations V(x) is the “potential” that describes the external force on a particle in a quantum system In general the wavefunction is a complex (i.e. involves imaginary ∣ x ,t ∣ 2 dx

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6 Schroedinger's equation: visual example. Schroedinger's equation: visual example. Time-dependent Schroedinger Equation: • Wavefunctions that determine the probability of finding a quantum particle undergoing simple harmonic motion. i ∂ t = −ℏ 2 2m 2 x 2 V x  Wavefunction is a complex: Red = imaginary part Blue = real part ∣ x ,t ∣ 2 dx C,D,E,F: Standing waves??
Complex numbers: a brief reference Complex numbers: a brief reference • Complex number has a real part and an imaginary part Think: vector components in two dimensions • Complex conjugation: change the sign of the imaginary part ( a + ib )* = a - ib Magnitude of a complex # |a+ib| 2 = ( a + ib )* ( a + ib ) = a 2 + b 2 Euler's relation: e i = cos

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## This document was uploaded on 10/27/2011 for the course PHYS 214 at Penn State.

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Lecture 4 - Particles of Light Lecture 7 Matter Waves...

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