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Chapter 3: Problem 11

Calculate the angle that the angular momentum vector makes with the \(z\) -axis when the system is described by the wavefunction \(\psi_{l m_{i}}\). Show that the minimum angle approaches zero as \(l\) approaches infinity. Calculate the allowed angles when \(l\) is \(1,2,\) and 3

Short Answer

Expert verified
The minimum angle \(\theta\) that the angular momentum vector makes with the \(z\)-axis approaches \(0\) as \(l\) approaches infinity. For \(l=1,2,3\), we must calculate the inverse cosine of the values obtained through \(cos(\theta_{min}) = \frac{l}{\sqrt{l(l+1)}}\).

Step by step solution

01

Define Angular Momentum Vector

In quantum mechanics, the angular momentum is given by \(\vec{L} = \vec{r} \times \vec{p}\), where \(\vec{r}\) is the radial vector and \(\vec{p}\) is the momentum vector. But for a particle in a central-field (like hydrogen atom), we use quantum numbers \(l\) and \(m\). The magnitude of angular momentum is given by \(\hbar \sqrt{l(l+1)}\) where \(\hbar\) is the reduced Planck's constant.
02

Calculate z-component of Angular Momentum

The z-component of the angular momentum for a quantum state with quantum numbers \(l\) and \(m\) is given by \(\hbar m\). Both \(l\) and \(m\) can take on integer values. \(m\) can range from \(-l\) to \(+l\). The vector direction is quantized in the z direction with the maximum z component is \(L_{z(max)}=\hbar l\).
03

Compute the Angle

The cosine of the angle \(\theta\) between the angular momentum vector and z-axis is given by \(cos(\theta) = \frac{L_z}{|\vec{L}|}\). Plugging in our values, we get \(cos(\theta) = \frac{\hbar m}{\hbar \sqrt{l(l+1)}} = \frac{m}{\sqrt{l(l+1)}}\). We’re asked to find the minimal angle, and that happens when \(m=l\), so we find \(cos(\theta_{min}) = \frac{l}{\sqrt{l(l+1)}}\). The smaller the angle \(\theta\), the larger \(cos(\theta)\) becomes. Thus \(cos(\theta_{min}) = 1\), meaning \(\theta_{min} = 0\) as \(l\) approaches infinity.
04

Calculating Angles for \(l = 1, 2, 3\)

Now we apply the above formula to calculate the minimum angle for each value: For \(l=1\), we find \(cos(\theta_{min}) = \frac{1}{\sqrt{1(1+1)}} = \frac{1}{\sqrt{2}}\). Therefore, \(\theta_{min} = cos^{-1} (\frac{1}{\sqrt{2}})\). Likewise, for \(l=2\) and \(l=3\), apply the same formula and calculate \(\theta_{min}\) by taking the inverse cosine of \(cos(\theta_{min})\).

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Most popular questions from this chapter

(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.

(a) Confirm that the radius of gyration of a solid uniform sphere of radius \(R\) is \(r=\left(\frac{2}{5}\right)^{1 / 2} R .\) (b) What is the radius of gyration of a solid uniform cylinder of radius \(R\) and length \(l\) about an axis perpendicular to its principal axis? Hint. For part (a), the moment of incrtia of a sphere is \(^{2} / s M R^{2}\)

Evaluate the probability that a particle of mass \(m\) in the ground state of a circular square well of radius \(a\) will be found within the circular area of radius \(1 / 2 a\)

Calculate (a) the mean radius, (b) the mean square radius, and (c) the most probable radius of the 1 s- \(, 2\) sand 3 s-orbitals of a hydrogenic atom of atomic number \(Z\) Hint. For the most probable radius look for the principal maximum of the radial distribution function. You will find the following integral useful: $$\int_{0}^{\infty} x^{n} \mathrm{c}^{-\alpha x} \mathrm{d} x=\frac{n !}{a^{n+1}}$$
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