(a) Construct a wavepacket \(\Psi=\mathrm{N} \sum_{m_{\mathrm{r}}=0}^{\infty}\left(1 / m_{l} !\right) \mathrm{e}^{\mathrm{in} \varphi}\) and normalize it to unity, Sketch the form of \(|\Psi|^{2}\) for \(0 \leq \varphi \leq 2 \pi\) (b) Calculate \(\langle\varphi\rangle,\langle\sin \varphi\rangle,\) and \(\left\langle l_{z}\right\rangle\) (c) Why is \(\left\langle l_{z}\right\rangle \leq \hbar ?\) Hint. Draw on a variety of pieces of information, including \(\sum_{n=0}^{\infty} x^{n} / n !=\mathrm{e}^{x}\) and the following integrals: $$\int_{0}^{2 \pi} \mathrm{e}^{\operatorname{scos} \phi} \mathrm{d} \varphi=2 \pi I_{0}(z) \int_{0}^{2 \pi} \cos \varphi \mathrm{e}^{\sec \varphi} \mathrm{d} \varphi=2 \pi I_{1}(z)$$ with \(I_{0}(2)=2.280 \ldots, I_{1}(2)=1.591 \ldots ;\) the \(I(z)\) are modificd Bessel functions.

Step by step solution

01

Construction and Normalization of the Wavepacket

First, construct the wave packet function \(\Psi\) using the given series including the normalization constant N. To find the normalization constant, set the integral from 0 to \(2\pi\) of \(|\Psi|^2\) equal to 1 and solve for N.

02

Sketch \(|\Psi|^2\)

To sketch the form of \(|\Psi|^2\), plot against \(\varphi\) for the range [0, \(2\pi\)]. Notice that values will be real and positive as it's a probability distribution. In this case, some knowledge of software for graphing functions will be beneficial.

03

Calculate \(\langle\varphi\rangle\), \(\langle\sin \varphi\rangle\) and \(\langle l_z \rangle\)

The expectation value of an operator is calculated by taking the integral of the conjugate of the wave function multiplied by the operator applied to the wave function, over all space. Use this formula to compute \(\langle\varphi\rangle\), \(\langle\sin \varphi\rangle\) and \(\langle l_z \rangle\). Remember to multiply by the normalization constant.

04

Evaluate \(\langle l_z \rangle\) with respect to \(\hbar\)

To understand why \(\langle l_z \rangle \leq \hbar\), it's important to note that \(l_z\) is the z component of the angular momentum and is quantized in units of \(\hbar\) - this is a fundamental postulate of quantum mechanics known as quantization of angular momentum. On physical grounds, because it's a component of the angular momentum, it must be less than or equal to the total angular momentum which is quantized and takes a minimum value of \(\hbar\).

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