ps01 - out your calculation by integrating every other x t...

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Physics 651 – Problem Set 1 (due 9/8/03) Read PS 9 . 1 , p.275 to p.282. 1. Evaluate the following ordinary integral as a function of K and J for general N : Z dx 1 dx 2 ...dx N e - x i K ij x j / 2+ x i J i (1) where repeated indices are summed. K is a real symmetric N × N constant matrix and J is a real constant N -component vector. (First consider the N = 1 and N = 2 cases. You may also set J = 0 first. Then go to the basis where K is diagonal. Finally consider the general case.) 2. The measure in functional integration in quantum mechanics. You will check that the measure given in the textbook reproduces the correct free particle propagator, which can be obtained in standard quantum mechanics. Use standard method, i.e., Eq.(9.1), to obtain the propagator U ( x F , x I , t F , t I ) for a free particle (derived in class). Use the definition of path integral (Eq.(9.3), (9.4), (9.6)) to obtain the same result. Here, use the “block spin” method to evaluate the functional integral: let the time interval be discretized into N = 2 n steps and then take the large N limit. Carry
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Unformatted text preview: out your calculation by integrating every other x ( t ). For example, for N = 8, first you integrate x 1 , x 3 , x 5 and x 7 , resulting in a new N = 4. Next, you integrate x 2 and x 6 , resulting in a new N = 2. Finally you integrate x 4 to obtain the final result. 3. Now generalize the path integral to the 3-dim case. Rederive the Schroedinger equation from the path integral formalism for the 3-dim case. 4. Show that the Lorentz generators L μν = i ( x μ ∂ ν-x ν ∂ μ ) form the Lie algebra of the Lorentz group. (That is, each of their commutators can be expressed as a linear combinations of L μν only.) Now include the translation generators P μ =-i∂ μ . Show that, together, they form the Lie algebra of the Poincare group. Restrcting μ to spatial indices only, show that J i = ± ijk L jk / 2 form the usual angular momentum algebra. That is, the rotation group is a subgroup of the Lorentz group. 1...
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This note was uploaded on 09/28/2008 for the course PHYS 651 taught by Professor Tye,henry during the Fall '03 term at Cornell.

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