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Chapter 1: Problem 19

(a) Show that \(\mathrm{e}^{A} \mathrm{e}^{B}=\mathrm{e}^{A+B}\) only if \([A, B]=0 .\) (b) If \([A, B] \neq 0\) but \([A,[A, B]]=[B,[A, B]]=0,\) show that \(\mathrm{e}^{A} \mathrm{e}^{B}=\) \(\mathrm{e}^{A+B} \mathrm{e}^{f},\) where \(f\) is a simple function of \([A, B] .\) Hint. This is another example of the differences between operators \((q-\text { numbers })\) and ordinary numbers (c-numbers). The simplest approach is to expand the exponentials and to collect and compare terms on both sides of the equality. Note that \(\mathrm{e}^{A} \mathrm{e}^{B}\) will give terms like \(2 A B\) while \(\mathrm{e}^{A+B}\) will give \(A B+B A .\) Be careful with order.

Short Answer

Expert verified
In conclusion, \(\mathrm{e}^{A} \mathrm{e}^{B}=\mathrm{e}^{A+B}\) if and only if the commutator \([A, B] = 0\). If \([A, B] \neq 0\) but \([A,[A, B]] = [B,[A, B]] = 0\), then \(\mathrm{e}^{A} \mathrm{e}^{B}=\mathrm{e}^{A+B} \mathrm{e}^{f}\), where \(f\) corresponds to the second order and higher order differences.

Step by step solution

01

Part (a) - Step 1: Expand the Exponentials

First, expand the exponentials using the Taylor Series Expansion. For the case of \(\mathrm{e}^{A} \mathrm{e}^{B}\), it will have terms like \(A B\) and for the case of \(\mathrm{e}^{A+B}\) it will have terms like \(A B + B A\). Abiding by commutative law of addition for operators, \(A B + B A = 2 A B\) if and only if \([A, B] = 0\) where \([A, B]\) is the commutator defined as \([A, B] = AB - BA\). Hence, if \([A, B] = 0\), \(A B = B A\) and \(\mathrm{e}^{A} \mathrm{e}^{B} =\mathrm{e}^{A+B}\).
02

Part (b) - Step 2: Considering the second condition

Now, for the case where \([A, B] \neq 0\) but \([A,[A, B]] = [B,[A, B]] = 0\), establish a series expansion for both \(\mathrm{e}^{A} \mathrm{e}^{B}\) and \(\mathrm{e}^{A+B}\) as done in the previous step. Here, however, terms of order two and above will include mixed terms. The differences can be expressed as an additional exponential function \(\mathrm{e}^{f}\), where \(f\) is a simple function corresponding to the second order and higher order terms of the difference.

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Most popular questions from this chapter

The operator \(e^{A}\) has a meaning if it is expanded as a power scrics: \(\mathrm{e}^{A}=\Sigma_{n}(1 / n !) A^{n} .\) Show that if \(|a\rangle\) is an eigenstate of \(A\) with eigenvalue \(a,\) then it is also an eigenstate of \(\mathrm{e}^{A} .\) Find the latter's eigenvalue.

A particle is moving in a circle in the \(x y\) plane. The only coordinate of importance is the angle \(\varphi\) which can vary from 0 to \(2 \pi\) as the particle goes around the circle. We are interested in measurements of the angular momentum \(l_{z}\) of the particle. The angular momentum operator for such a system is given by \((\hbar / \mathrm{i}) \mathrm{d} / \mathrm{d} \varphi\) (a) Suppose that the state of the particle is described by the wavefunction \(\psi(\varphi)=N \mathrm{e}^{-\mathrm{i} \varphi}\) where \(N\) is the normalization constant. What values will we find when we measure the angular momentum of the particle? If more than one value is possible, what is the probability of obtaining each result? What is the expectation value of the angular momentum? (b) Now suppose that the state of the particle is described by the normalized wavefunction \(\psi(\varphi)=N\left\\{(3 / 4)^{1 / 2} \mathrm{e}^{-i \varphi}-(\mathrm{i} / 2) \mathrm{e}^{2 \text { ip }}\right\\} .\) When we measure the angular momentum of the particle, what values will we find? If more than one value is possible, what is the probability of obtaining each result? What is the expectation value of the angular momentum?

Evaluate the commutators (a) \(\left[(1 / x), p_{x}\right]\) (b) \(\left[(1 / x), p_{x}^{2}\right]\) (c) \(\left[x p_{y}-y p_{x}, y p_{z}-z p_{y}\right]\) (d) \(\left[x^{2}\left(\partial^{2} / \partial y^{2}\right), y(\partial / \partial x)\right]\)

Evaluate the commutators (a) \(\left[H, p_{x}\right]\) and (b) \([H, x]\) where \(H=p_{x}^{2} / 2 m+V(x) .\) Choose (i) \(V(x)=V,\) a constant, (ii) \(V(x)=\frac{1}{2} k_{t} x^{2},\) (iii) \(V(x) \rightarrow V(r)=e^{2} / 4 \pi \varepsilon_{0} r .\) Hint. For part (b), case (iii), use \(\left(\partial r^{-1} / \partial x\right)=-x / r^{3}\)

Are the linear combinations \(2 x-y-z, 2 y-x-z\) \(2 z-x-y\) linearly independent?
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