Lecture34

# Lecture34 - Physics 344 Foundations of 21st Century Physics...

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Physics 344 Foundations of 21 st Century Physics: Relativistic and Quantum Systems Instructor: Dr. Mark Haugan Office: PHYS 282 TA: Dan Hartzler Office: PHYS 7 Grader: Fan Chen Office: PHYS 222 Office Hours: If you have questions, just email us to make an appointment. We enjoy talking about physics! Help Session: Thursdays 2:00 – 4:00 in PHYS 154 Reading: Chapters 10 and 11 in the text. Exam 2: Thursday, December 1 at 8:00pm in ARMS B061

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Idealizing the Electron-Metal Film System We modeled the system consisting of a conduction electron interacting with the rest of a conducting metal film in which it is bound using a square potential well with depth determined by the workfunction of the conductor. 0, 0 ( ) , 0 0, x V x W x a x a < = - < < The finite binding of this model allowed us to demonstrate that it is the condition that bound energy eigenstates of a system be normalizable that leads to the discrete energy spectrum characteristic of quantum bound states. For a solution of the time-independent Schrödinger equation to represent a state 2 2 2 ( ) ( ) ( ) 2 E E E d E x V x x m dx ψ = - + the wavefunction Ψ E must fall off rapidly enough far from the well for * ( ) ( ) finite value E E x x dx -∞ = We saw why this is only possible for certain discrete values of the energy E .
Our finite-square potential well model also revealed the distinctly quantum mechanical phenomenon of tunneling. The probability of detecting the electron in a classically forbidden region outside the metal film is nonzero. Units in this figure put the film’s edges at x = 0 and 1. 0 0 * 3 3 ( ) ( ) ( ) 0 x x dx x dx ψ ρ -∞ -∞ We’ve used the index 3 to label this state because the corresponding plot of the time-independent ρ (x) has three “bumps”. This case illustrates a general rule that the more bumps an energy eigenvector’s probability density has the larger its energy eigenvalue is. The more rapidly these energy eigenfunctions oscillate the greater the contribution their kinetic energy expectation value, 3 ( ) x The relationship between a statevector and the probability density ρ (x) means that the probability of detecting the electron to the left of the well when the electron-film system is in the state Ψ 3 shown is 2 2 * 3 3 3 2 ( ) ( ) 2 K x x dx m x -∞ = - makes to the energy eigenvalue E 3 = < K 3 > + < V 3 >.

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Lecture34 - Physics 344 Foundations of 21st Century Physics...

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