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# HW4 - e-y 2 H n y H m y(c Using these results compute the...

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Physics 731 Assignment #4, due Monday, October 5 1. Determine the (normalized) odd parity eigenfunctions and allowed energies for a particle bound in the ﬁnite square well: V = 0 , | x | < a V 0 , | x | > a, in which V 0 > 0 . Discuss the limiting behavior as V 0 0 and V 0 → ∞ . 2. Find accurate numerical values for the bound state energy eigenvalues of a particle in the above ﬁnite square well potential, in which ± 2 mV 0 a 2 ¯ h 2 ! 1 2 = 2 . Do this (a) numerically, and (b) graphically (with reasonable precision). 3. (a) Use the Hermite generating function, g ( y,t ) = e - t 2 +2 ty = X n =0 t n n ! H n ( y ) , to prove the following expressions: H n ( y ) = e y 2 / 2 ² y - d dy ³ n e - y 2 / 2 H 0 n ( y ) = 2 nH n - 1 ( y ) H n +1 ( y ) = 2 yH n ( y ) - 2 nH n - 1 ( y ) , (b) and to evaluate Z -∞
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Unformatted text preview: e-y 2 H n ( y ) H m ( y ) . (c) Using these results, compute the matrix elements of X and P between two energy eigenfunctions of the one-dimensional simple harmonic oscillator: h n | X | m i and h n | P | m i . 4. Calculate the probability that a particle in the ground state of the one-dimensional simple harmonic oscillator is farther from the origin than the classical turning points (where E = V ). 5. Evaluate both sides of the uncertainty relation for the n th energy eigenstate of the one-dimensional simple harmonic oscillator. 1...
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