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1 Practice Self Evaluation #3 (Chem113A W’07, Thu. 03/22/07 11:30a-2:30p, 100+5pts, max=100pts) Name: ____ [1] Rotational and vibrational spectroscopy [1](a) Diatomic vibrational spectroscopy. [1](b) Diatomic rotational spectroscopy. [1](c) Ro-vibrational coupling. [1](d) Relative population between rotational states by Boltzmann distribution The relative population between two states i and j are: Population,i/Population,j = (g i /g j ) exp[-(E i -E j )/(k B T )] where g i and g j are the degeneracy of these two states and E i and E j are the energies of these two states. Please solve the following question.
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2 [2] Orbital angular momentum: from classical mechanics to quantum mechanics [2](a) Orbital angular momentum in CM. QM originates from CM. So let’s see what CM says about orbital angular momentum. In classical mechanics (CM), angular momentum L (a vector) is defined as L = r × p where vector L = L x i + L y j + L z k , vector r = x i + y j + z k , and vector p = p x i + p y j + p z k . Please show by algebra that the x, y, z components of L are: L x = yp z - zp y ; L y = zp x - xp z ; L z = xp y yp x . [2](b) Orbital angular momentum in QM. After promoting all the dynamical variables into operators, one reaches the corresponding expression in QM. But there is a catch: all QM operators correspond to experimental observables must be Hermitian. Please show by adjoint operator algebra that L z = xp y yp x (bold means QM operator) is indeed a Hermitian operator. [2](c) Real eigenvalues theorem. One of the reasons that all QM operators correspond to experimental observables must be Hermitian is that the eigenvalues of a Hermitian operator are all real. That is: for eigenvalue equation O | Ψ > = λ | Ψ >, if O + = O then λ is real. Please prove this theorem by algebra (Hint: dual- space correspondence) [2](d) Commutator relations. Given the formula in [2](a) for L x , L y and L z and the commutator [ x , p x ] = i (in reduced units), please show by commutator algebra that [ L x , L y ] = i L z (in reduced units). [2](e) Common eigenkets. Similarly, one can show that, in reduced units, [ L y , L z ] = i L x and [ L z , L x ] = i L y (no need to prove). Based on these commutators, please calculate the commutator [ L 2 , L z ] by commutator algebra. (Remark: this result suggests that QM operators L 2 and L z can share a set of common eigenkets) [2](f) Expectation values. Denote the common eigenkets of L 2 and L z by | l,m > where l and m are two quantum numbers used to label the eigenkets (in reduced units):
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