Hw0 - r = 10-8 cm This instability was one of the reasons for the discovery of quantum mechanics 2 The Pauli matrices are defined by σ 1 = ± 0 1

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HOMEWORK # 0 Physics 6572 Friday, 9/5/08 Do not turn in this homework. The purpose of this assignment is for you to review your background knowledge that is relevant to this course: failure of classical physics, manipulation of simple ma- trices, and method of Fourier transforms. Try to do as much as possible by yourself before seeking help. makes it unstable for an electron of charge e and mass m to move in a circular orbit under the Coulomb forces produced by the proton. Suppose as it radiates, the electron continues to move on a circle. In a finite time the electron reaches the center. Calculate how long it takes the electron to hit the proton if it starts from an initial radius
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Unformatted text preview: r = 10-8 cm . This instability was one of the reasons for the discovery of quantum mechanics. 2. The Pauli matrices are defined by σ 1 = ± 0 1 1 0 ² , σ 2 = ±-i i ² , σ 3 = ± 1-1 ² . Let us consider the three matrices as components of a vector ~σ . Show that a) e i ˆ n · ~σφ = cos φ + i ˆ n · ~σ sin φ , where ˆ n is a unit vector. b) ( ~a · ~σ ) ( ~ b · ~σ ) = ~a · ~ b + i~σ · ~a × ~ b , where ~a and ~ b are any two ordinary vectors. 3. The Fourier transform of a function ψ ( ~ r ) is defined by the relation: φ ( ~ p ) = Z d 3 r (2 π ¯ h ) 3 / 2 e-i ¯ h ~ p · ~ r ψ ( ~ r ) . Find the Fourier transforms of the two functions ψ 1 ( ~ r ) = 1 | ~ r | ψ 2 ( ~ r ) = e-a 2 ~ r 2 ....
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This note was uploaded on 03/27/2011 for the course PHYS 6572 at Cornell University (Engineering School).

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