Chapter 8: Spin and Atomic Physics

Exercise 44 gives an antisymmetric multiparticle state for two particles in a box with opposite spins. Another antisymmetric state with spins opposite and the same quantum numbers is ${\mathbf{\psi }}_{\mathbf{n}}\mathbf{\left(}{\mathbf{x}}_{\mathbf{1}}\mathbf{\right)}{\mathbf{\downarrow }}_{{\mathbf{n}}^{\mathbf{2}}}\mathbf{\left(}{\mathbf{x}}_{\mathbf{2}}\mathbf{\right)}\mathbf{\uparrow }\mathbf{-}{\mathbf{\psi }}_{{\mathbf{n}}^{\mathbf{n}}}\mathbf{\left(}{\mathbf{x}}_{\mathbf{1}}\mathbf{\right)}\mathbf{\uparrow }{\mathbf{\psi }}_{\mathbf{n}}\mathbf{\left(}{\mathbf{x}}_{\mathbf{2}}\mathbf{\right)}\mathbf{\downarrow }$

Refer to these states as 1 and 11. We have tended to characterize exchange symmetry as to whether the state's sign changes when we swap particle labels. but we could achieve the same result by instead swapping the particles' stares, specifically theandin equation (8-22). In this exercise. we look at swapping only parts of the state-spatial or spin.

(a) What is the exchange symmetric-symmetric (unchanged). antisymmetric (switching sign). or neither-of multiparticle states 1 and Itwith respect to swapping spatial states alone?

(b) Answer the same question. but with respect to swapping spin states/arrows alone.

(c) Show that the algebraic sum of states I and II may be written$\mathbf{(}{\mathbf{\psi }}_{\mathbf{n}}\mathbf{\left(}{\mathbf{x}}_{\mathbf{1}}\mathbf{\right)}{\mathbf{\psi }}_{{\mathbf{n}}^{\mathbf{\text{'}}}}\mathbf{\left(}{\mathbf{x}}_{\mathbf{2}}\mathbf{\right)}\mathbf{-}{\mathbf{\psi }}_{{\mathbf{n}}^{\mathbf{\text{'}}}}\mathbf{\left(}{\mathbf{x}}_{\mathbf{1}}\mathbf{\right)}{\mathbf{\psi }}_{\mathbf{n}}\mathbf{\left(}{\mathbf{x}}_{\mathbf{2}}\mathbf{\right)}\mathbf{)}\mathbf{(}\mathbf{\downarrow }\mathbf{\uparrow }\mathbf{+}\mathbf{\uparrow }\mathbf{\downarrow }\mathbf{)}$

Where the left arrow in any couple represents the spin of particle 1 and the right arrow that of particle?

(d) Answer the same questions as in parts (a) and (b), but for this algebraic sum.

(e) ls the sum of states I and 11 still antisymmetric if we swap the particles' total-spatial plus spin-states?

(f) if the two particles repel each other, would any of the three multiparticle states-l. II. and the sum-be preferred?

Explain.

Exercise 45 refers to state I and II and put their algebraic sum in a simple form. (a) Determine algebraic difference of state I and state II.

(b) Determine whether after swapping spatial state and spin state separately, the algebraic difference of state I and state II is symmetric, antisymmetric or neither, and to check whether the algebraic difference becomes antisymmetric after swapping spatial and spin states both.

A lithium atom has three electrons. These occupy individual particle states corresponding to the sets of four quantum numbers given by .

$\left(n,l,m_l,m_j\right)\mathbf{=}\left(1,0,0,+\frac{1}{2}\right)\mathbf{,}\left(1,0,0,-\frac{1}{2}\right)\mathbf{}\mathbf{\text{and}}\mathbf{}\left(2,0,0,+\frac{1}{2}\right)$

Using ${\mathit{\psi }}_{\mathbf{1}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{0}}\left(r_j\right)\mathbf{\uparrow }\mathbf{,}{\mathit{\psi }}_{\mathbf{1}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{0}}\left(r_j\right)\mathbf{\downarrow }\mathbf{,}\mathbf{}\mathbf{\text{and}}\mathbf{}{\mathit{\psi }}_{\mathbf{2}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{0}}\left(r_j\right)\mathbf{\uparrow }$ to represent the individual-particle states when occupied by particle$j$ . Apply the Slater determinant discussed in Exercise 42 to find an expression for an antisymmetric multiparticle state. Your answer should be sums of terms like .

${\mathit{\psi }}_{\mathbf{1}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{0}}\left(r_1\right)\mathbf{\uparrow }\mathbf{,}{\mathit{\psi }}_{\mathbf{1}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{0}}\left(r_2\right)\mathbf{\downarrow }\mathbf{,}\mathbf{}\mathbf{\text{and}}\mathbf{}{\mathit{\psi }}_{\mathbf{2}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{0}}\left(r_3\right)\mathbf{\uparrow }$

To investigate the claim that lowerimplies lower f energy. consider a simple case: lithium. which has two$\mathit{n}\mathbf{=}\mathbf{1}$electrons and alone$\mathit{n}\mathbf{=}\mathbf{2}$valence electron.

(a)First find the approximate orbit radius, in terms of${\mathit{a}}_{\mathbf{0}}$. of an$\mathit{n}\mathbf{=}\mathbf{1}$electron orbiting three protons. (Refer to Section 7.8.)

(b) Assuming the$\mathit{n}\mathbf{=}\mathbf{1}$electrons shield/cancel out two of the protons in lithium's nucleus, the orbit radius of an$\mathit{n}\mathbf{=}\mathbf{2}$electron orbiting a net charge of just$\mathbf{+}\mathit{e}$.

(c) Argue that lithium's valence electron should certainly have lower energy in a 25 state than in a$\mathbf{2}\mathit{p}$stale. (Refer Figure 7.15.)

Determine the electronic configuration for phosphorus, germanium and cesium.

Compare and contrast the angular momentum and magnetic moment related to orbital motion with those that are intrinsic.

Determine the expected valence of the element with atomic number 117.

Were it to follow the standard pattern, what would be the electronic configuration of element 119.

The radius of cesium is roughly$\mathbf{0}\mathbf{.}\mathbf{26}\mathbf{}\mathbf{\text{nm}}$.

(a) From this estimate the effective charge its valence electron orbits

(b) Given the nature of the electron's orbit. is this effective nuclearcharge reasonable?

(c) Compare this effective Zwith that obtained for sodium in Example 8.3. Are the values at odds with the evidence given in Figure$\mathbf{8}\mathbf{.}\mathbf{16}$that it takes less energy to remove an electron from cesium than from sodium? Explain.

Consider row 4 of the periodic table. The trend is that the$\mathbf{4}\mathit{x}$subshell fills. Then the 3d, then the 4p.

(a) Judging by adherence to and deviation from this trend, whit might be said of the energy difference between the 4sand 3drelative to that between the 3dand 4p?

(b) Is this also true of row 5?

(c) Are these observations in qualitative agreement with Figure 8.13? Explain.