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Chapter 12: Problem 10

Prove the sum rule \\[ \Sigma_{p} x_{m f} x_{f m} \omega_{f n}=\left(\hbar / 2 m_{e}\right) \delta_{m n}+\frac{1}{2} \omega_{m n}\left(x^{2}\right)_{m n^{+}} \\] Hint. Consider the matrix elements of the commutator \\[ \left[H, x^{2}\right] \\].

Short Answer

Expert verified
While the problem is complex involving multiple concepts in quantum mechanics, the main steps involve calculating the left-hand side and right-hand side of the equation separately and then proving their equality. An important part of the proof involves calculating the matrix elements of the commutator \([H, x^{2}]\).

Step by step solution

01

Identify the problem

The problem is to prove the equation given. This will involve calculating the left-hand side of the equation, which is the sum rule with operators, and comparing it with the right side to confirm they are indeed equal.
02

Calculate the Left-hand side

The left-hand side of the equation can be written as follows using summation notation: \[ \Sigma_{p} x_{m f} x_{f m} \omega_{f n} \]. It represents a sum over \(p\), of the product of the quantum states \(x_{m f}\), \(x_{f m}\) and the frequency \(\omega_{f n}\). Calculate this sum.
03

Calculate the Right-hand side

The right-hand side of the equation is: \[ \left(\hbar / 2 m_{e}\right) \delta_{m n}+\frac{1}{2} \omega_{m n}\left(x^{2}\right)_{m n^{+}} \]. The first term is the product of the reduced Planck's constant \(\hbar\), divided by twice the electron mass \(m_e\), and the Kronecker delta \(\delta_{m n}\). The second term is half the product of the frequency \(\omega_{m n}\) and the square of the position operator \(x\) in bra-ket notation. Calculate this expression.
04

Compute the commutator

The hint suggests considering the matrix elements of the commutator \[ \left[H, x^{2}\right] \], where \(H\) is the Hamiltonian operator and \(x^{2}\) is the square of the position operator. A commutator in quantum mechanics is defined as \[ [A,B] = AB - BA \], so calculate \[ [H, x^{2}] \].
05

Prove the identity

The final step is to show that the left hand side and right hand side of the equation are indeed equal, hence proving the identity. This should involve the results from the computations made in steps 2, 3, and 4.

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Most popular questions from this chapter

Evaluate the polarizability \(\alpha_{z z}\) and polarizability volume of a hydrogen atom; for simplicity, confine the perturbation sum to the 2 p-orbitals. Use the following results for matrix elements: \\[ \begin{array}{l} \left\langle n^{\prime}, l+1, m_{l}|z| n l m_{l}\right\rangle=\left\\{\frac{(l+1)^{2}-m_{l}^{2}}{(2 l+3)(2 l+1)}\right\\}^{1 / 2}\left\langle n^{\prime}, l+1|r| n l\right\rangle \\ \langle n p|r| 1 \mathrm{s}\rangle=\left\\{\frac{2^{8} n^{7}(n-1)^{2 n-5}}{(n+1)^{2 n+5}}\right\\}^{1 / 2} a_{0} \end{array} \\]

Evaluate the rotational strength of a transition of an electron from a \(2 \mathrm{p}_{x}\) -orbital to a \(2 \mathrm{p}_{z}, 3 \mathrm{d}_{x y}\) -hybrid orbital. Assume the orbitals are on a carbon atom. Estimate the optical rotation angle for 590 nm light. Hint. Follow Example \(12.4,\) with changes of detail. For carbon, take \(\zeta_{p}=1.57 / a_{0}\) and \(\zeta_{d}=0.33 / a_{0}\) and use \(\lambda_{k 0}=200 \mathrm{nm}\).

Show group theoretically that in a tetrahedral molecule (a) the mean hyperpolarizability is zero, (b) the only non-zero components are \(\beta_{x y z}\) and the permutations of its indices. Hint. The mean is defined as \(\frac{3}{3}\left(\beta_{x x z}+\beta_{y z z}+\beta_{z z z}\right)\) and so (b) implies (a). For (b) consider the symmetry characteristics of \(E=-(1 / 3 !) \Sigma_{a, b, c} \beta_{a b c} E_{a} E_{b} E_{c}\) the generalization of eqn 12.11.
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