3502_11_octo04

3502_11_octo04 - Chem 3502/5502 Physical Chemistry II...

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Chem 3502/5502 Physical Chemistry II (Quantum Mechanics) 3 Credits Fall Semester 2010 Laura Gagliardi Lecture 11, October 4, 2010 Solved Homework We are asked to find < T > for the first harmonic oscillator wave function Ψ 0 x ( ) = μ π 1/ 4 e μ x 2 / 2 So, for < T > n =0 we have T n = 0 = k μ π e μ x 2 /2 2 2 μ d 2 dx 2 −∞ μ π e μ x 2 dx We need to evaluate the second derivative. It is d 2 dx 2 e μ x 2 / 2 = d dx d dx e μ x 2 / 2 = d dx x μ e μ x 2 / 2 = μ e μ x 2 / 2 + x 2 k μ 2 e μ x 2 / 2 Thus, T n = 0 = 2 2 μ k μ π 1/ 2 k μ 2 x 2 e μ x 2 / −∞ dx k μ e μ x 2 / −∞ dx We've already evaluated the two integrals in previous work. Using the appropriate formulae provides
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11-2 T n = 0 = 2 2 μ μ π 1/ 2 k μ 2 2 μ π μ 1 /2 μ π μ = 2 2 μ μ π μ 2 π μ = 4 k μ Recall from our last homework that x 2 n = 0 = 2 μ The expectation value of the potential energy < V > is simply ( k /2)< x 2 >, or 1 2 k x 2 n = 0 = k 2 2 μ = 4 k μ This completes the proof that < T > = < V > for the n = 0 state of the QMHO. It is similarly straightforward, if increasingly tedious, to prove this for the first excited state, and indeed for any state. Angular Momentum Angular momentum is a vector quantity, defined as the cross product of the position vector and the momentum vector. In cartesian coordinates, it is most easily expressed as the determinant L = i j k x y z p x p y p z (11-1) where x , y , and z are the components of the position vector (i.e., the coefficients multiplying the unit vectors i , j , and k , respectively) and p x , p y , and p z are the
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11-3 components of the momentum vector. A 3 x 3 determinant may be evaluated by Cramer's rule (no relation to your enthusiastic instructor, as far as I know. ..), which states that the determinant is equal to the sum of the three down-right wraparound multiplications minus the sum of the three up-right wraparound multiplications. That is i j k x y z p x p y p z = yp z i + zp x j + xp y k yp x k zp y i xp z j = yp z zp y ( ) i + zp x xp z ( ) j + xp y yp x ( ) k (11-2) Thus, the components of L , namely, L x , L y , and L z , are the terms in parentheses preceding the corresponding unit vectors. In the absence of a torque on a system, angular momentum is a conserved quantity, just as linear momentum is conserved in the absence of a force on a system.
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This note was uploaded on 12/18/2010 for the course CHEM 3502 taught by Professor Staff during the Fall '08 term at Minnesota.

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3502_11_octo04 - Chem 3502/5502 Physical Chemistry II...

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