Chapter 7: Quantum Mechanics in Three Dimensions and the Hydrogen Atom

A particle orbiting due to an attractive central force has angular momentum $\mathrm{L}=1.00\times {10}^{-33}\mathrm{kg}.\mathrm{m}/\mathrm{s}$ What z-components of angular momentum is it possible to detect?

In section 7.5,${\mathbf{e}}^{{\mathbf{im}}_{\mathbf{l}}\mathbf{\phi }}$is presented a sour preferred solution to the azimuthal equation, but there is more general one that need not violate the smoothness condition, and that in fact covers not only complex exponentials but also suitable redelinitions of multiplicative constants, sine, and cosine,

$\mathit{\Phi }{\mathit{m}}_{\mathbf{1}}\left(\Phi \right)\mathbf{=}\mathit{A}{\mathbf{e}}^{\mathbf{+}{\mathbf{im}}_{\mathbf{l}}\mathbf{\phi }}\mathbf{+}{\mathbf{Be}}^{\mathbf{+}{\mathbf{im}}_{\mathbf{l}}\mathbf{\phi }}$

(a) Show that the complex square of this function is not, in general, independent of $\mathbf{\phi }$.

(b) What conditions must be met by A and/or B for the probability density to be rotationally symmetric – that is, independent of $\mathbf{\phi }$ ?

Consider a 2D infinite well whose sides are of unequal length.

(a) Sketch the probability density as density of shading for the ground state.

(b) There are two likely choices for the next lowest energy. Sketch the probability density and explain how you know that this must be the next lowest energy. (Focus on the qualitative idea, avoiding unnecessary reference to calculations.)

Here we Pursue the more rigorous approach to the claim that the property quantized according to m_l is L_z,

(a) Starting with a straightforward application of the chain rule,

$\frac{\mathbf{\partial }}{\mathbf{\partial }\mathbf{\phi }\mathbf{\hspace{0.33em}}}\mathbf{=}\mathbf{\hspace{0.33em}}\frac{\mathbf{\partial }\mathbf{x}}{\mathbf{\partial }\mathbf{\phi }}\mathbf{/}\frac{\mathbf{}\mathbf{\partial }}{\mathbf{\partial }\mathbf{x}}\mathbf{+}\frac{\mathbf{\partial }\mathbf{y}}{\mathbf{\partial }\mathbf{\phi }}\mathbf{}\mathbf{}\frac{\mathbf{\partial }}{\mathbf{\partial }\mathbf{y}}\mathbf{+}\frac{\mathbf{\partial }\mathbf{z}}{\mathbf{\partial }\mathbf{\phi }}\mathbf{}\mathbf{}\frac{\mathbf{\partial }}{\mathbf{\partial }\mathbf{z}}$

Use the transformations given in Table 7.2 to show that

$\frac{\mathbf{\partial }}{\mathbf{\partial }\mathbf{\phi }}\mathbf{\hspace{0.33em}}\mathbf{=}\mathbf{\hspace{0.33em}}\mathbf{-}\mathit{y}\mathbf{}\frac{\mathbf{\partial }}{\mathbf{\partial }\mathbf{x}}\mathbf{+}\mathit{x}\frac{\mathbf{}\mathbf{\partial }}{\mathbf{\partial }\mathbf{y}}$

(b) Recall that L = r x p. From the z-component of this famous formula and the definition of operators for p_x and p_y, argue that the operator for L_z is $\mathbf{-}\mathit{i}\mathit{h}\mathbf{}\frac{\mathbf{\partial }}{\mathbf{\partial }\mathbf{\phi }}\mathbf{.}$.

(c) What now allows us to say that our azimuthal solution${\mathit{e}}^{\mathbf{i}{\mathbf{m}}_{\mathbf{l}}\mathbf{\phi }}$ has a well-defined z-component of angular momentum and that is value m_lh.

A simplified approach to the question of how lis related to angular momentum – due to P. W. Milonni and Richard Feynman – can be stated as follows: If can take on only those values ${\mathit{m}}_{\mathbf{l}}\mathit{h}$, where${\mathit{m}}_{\mathbf{l}}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{\pm }\mathbf{1}\mathbf{,}\mathbf{\dots }\mathbf{\dots }\mathbf{\pm }\mathit{l}$ , then its square is allowed only values${{\mathit{m}}_{\mathbf{l}}}^{\mathbf{2}}{\mathit{h}}^{\mathbf{2}}$, and the average of localid="1659178449093" ${\mathit{l}}^{\mathbf{2}}$should be the sum of its allowed values divided by the number of values,$\mathbf{2}\mathit{l}\mathbf{+}\mathbf{1}$ , because there really is no preferred direction in space, the averages of $\mathit{L}{\mathit{x}}^{\mathbf{2}}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{L}{\mathit{y}}^{\mathbf{2}}$should be the same, and sum of all three should give the average of role="math" localid="1659178641655" ${\mathit{L}}^{\mathbf{2}}$. Given the sumrole="math" localid="1659178770040" $\mathbf{\sum }_{\mathbf{1}}^{\mathbf{S}}{\mathit{n}}^{\mathbf{2}}\mathbf{=}\mathit{N}\mathbf{(}\mathit{N}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{(}\mathbf{2}\mathit{N}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{/}\mathbf{6}$, show that these arguments, the average of ${\mathit{L}}^{\mathbf{2}}$ should be $\mathit{l}\mathbf{(}\mathit{l}\mathbf{+}\mathbf{1}\mathbf{)}{\mathit{h}}^{\mathbf{2}}$.

In Appendix G. the operator for the square of the angular momentum is shown to be

${\hat{\mathbf{L}}}^{\mathbf{2}}\mathbf{=}\mathbf{-}{\sout{\mathbf{h}}}^{\mathbf{2}}\left[csc\theta \frac{\partial }{\partial \theta }\left(sin\theta \frac{\partial }{\partial \theta }\right)+cs{c}^2\theta \frac{{\partial }^2}{\partial {\varphi }^2}\right]$

Use this to rewrite equation (7-19) as${\hat{\mathbf{L}}}^{\mathbf{2}}\mathbf{⚇}\mathit{\Phi }\mathbf{=}\mathbf{-}\mathit{C}{\sout{\mathbf{h}}}^{\mathbf{2}}\mathbf{⚇}\mathit{\Phi }$

Explicitly verify that the simple function $R\left(r\right)=A{e}^{br}$can be made to satisfy radial equation (7-31), and in so doing, demonstrate what its angular momentum and energy must be.

How many different 3d states are there? What physical property (as opposed to quantum number) distinguishes them, and what different values may this property assume?

In Table 7.5, the pattern that develops with increasing n suggests that the number of different sets of$\left(\mathcal{l},m_{\mathcal{l}}\right)$values for a given energy level n is$n^2$. Prove this mathematically by summing the allowed values of$m_{\mathcal{l}}$for a given$\mathcal{l}$over the allowed values of$\mathcal{l}$for a given n.

Question: Show that the angular normalization constant in Table 7.3 for the case $\mathbf{(}\mathbf{l}\mathbf{,}{\mathbf{m}}_{\mathbf{l}}\mathbf{)}\mathbf{=}\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{0}\mathbf{)}$ is correct.