91Ó°ÊÓ

For a two-electron system, there are four possible spin functions: 1\. \(\alpha(1) \alpha(2)\) 2\. \(\beta(1) \alpha(2)\) 3\. \(\alpha(1) \beta(2)\) 4\. \(\beta(1) \beta(2)\) The concept of indistinguishability forces us to consider only linear combinations of 2 and 3 \\[ \psi_{\pm}=\frac{1}{\sqrt{2}}[\alpha(1) \beta(2) \pm \beta(1) \alpha(2)] \\] instead of 2 and 3 separately. Show that of the four acceptable spin functions, \(1,4,\) and \(\psi_{\pm}\) three are symmetric and one is antisymmetric. Now for a two-electron system, we can combine spatial wave functions with spin functions. Show that this combination leads to only four allowable combinations: \\[ \begin{array}{c} {[\psi(1) \phi(2)+\psi(2) \phi(1)] \frac{1}{\sqrt{2}}[\alpha(1) \beta(2)-\alpha(2) \beta(1)]} \\ {[\psi(1) \phi(2)-\psi(2) \phi(1)][\alpha(1) \alpha(2)]} \\ {[\psi(1) \phi(2)-\psi(2) \phi(1)][\beta(1) \beta(2)]} \end{array} \\] and \\[ [\psi(1) \phi(2)-\psi(2) \phi(1)] \frac{1}{\sqrt{2}}[\alpha(1) \beta(2)+\alpha(2) \beta(1)] \\] where \(\psi\) and \(\phi\) are two spatial wave functions. Show that \(M_{s}=m_{s 1}+m_{n}=0\) for the first of these and that \(M_{s}=1,-1,\) and 0 (in atomic units) for the next three, respectively. Consider the first excited state of a helium atom, in which \(\psi=1 s\) and \(\phi=2 s .\) The first of the four wave functions above, with the symmetric spatial part, will give a higher energy than the remaining three, which form a degenerate set of three. The first state is a singlet state and the degenerate set of three represents a triplet state. Because \(M,\) equals zero and only zero for the singlet state, the singlet state corresponds to \(S=0 .\) The other three, with \(M_{n}=\pm 1,0,\) corresponds to \(S=1 .\) Note that the degeneracy is \(2 S+1\) in each case. Putting all this information into a more mathematical form, given that \(\hat{S}_{\text {bat }}=\hat{s}_{1}+\hat{s}_{2}\) we can show that (Problem \(8-53\) ) \\[ \hat{S}_{\text {vatel }}^{2}[\alpha(1) \beta(2)-\alpha(2) \beta(1)]=0 \\] corresponding to \(S=0,\) and that \\[ \hat{S}_{\text {wal }}^{2}\left[\begin{array}{c} \alpha(1) \alpha(2) \\ \frac{1}{\sqrt{2}}[\alpha(1) \beta(2)+\alpha(2) \beta(1)] \\ \beta(1) \beta(2) \end{array}\right]=2 h^{2}\left[\begin{array}{c} \alpha(1) \alpha(2) \\ \frac{1}{\sqrt{2}}[\alpha(1) \beta(2)+\alpha(2) \beta(1)] \\ \beta(1) \beta(2) \end{array}\right] \\] corresponding to \(S=1\)

Step by step solution

01

Identify symmetry properties of spin functions

The spin functions are symmetric if swapping electrons 1 and 2 results in the same function, and antisymmetric if it results in a change of sign. For 1: \( \alpha(1) \alpha(2) \) and 4: \( \beta(1) \beta(2) \), if we swap particles, the function remains the same, so both are symmetric. \( \psi_+ = \frac{1}{\sqrt{2}}[\alpha(1)\beta(2) + \beta(1)\alpha(2)] \) is symmetric because exchange doesn't change the sign, while \( \psi_- = \frac{1}{\sqrt{2}}[\alpha(1)\beta(2) - \beta(1)\alpha(2)] \) is antisymmetric, as the sign changes.

02

Determine allowable combinations of wave functions

Upon forming wave functions, we must consider both symmetry in spatial and spin parts. The combination \([\psi(1) \phi(2) + \psi(2) \phi(1)] \frac{1}{\sqrt{2}}[\alpha(1)\beta(2) - \beta(1)\alpha(2)]\) is entirely antisymmetric, allowing the first wave function. The other three, \([\psi(1) \phi(2) - \psi(2) \phi(1)]\), combined with symmetric spin parts, form the other allowable combinations.

03

Analyze total spin projections \(M_s\)

The first combination involves a spin configuration \( \psi_- \), which has a total spin projection \(M_s = 0\). The other combinations paired with \(\alpha(1) \alpha(2)\), \(\beta(1) \beta(2)\), and \(\psi_+\) each result in projections of \(M_s = 1\), \(M_s = -1\), and \(M_s = 0\), respectively.

04

Discuss implications for helium atom states

For a helium system with \(\psi = 1s\) and \(\phi = 2s\), a symmetric spatial function corresponds to higher energy and a singlet state \(S=0\), due to degeneracy \(2S + 1 = 1\). In contrast, the antisymmetric spatial forms correspond to triplet states \(S=1\), with a degeneracy of 3, confirming the lowering of energy.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Spin Symmetry

In quantum mechanics, spin symmetry refers to the behavior of a wave function when the particles it describes are swapped. Two-electron systems can exhibit different types of spin symmetry: symmetric or antisymmetric. When the wave function remains unchanged upon the exchange of electrons, it is said to be symmetric. Conversely, if the wave function's sign changes, it is antisymmetric.

For instance, in a two-electron system:

• The functions \[\alpha(1) \alpha(2)\quad \text{and}\quad \beta(1) \beta(2)\]are symmetric because swapping electrons 1 and 2 does not alter these functions.
• The function \[\frac{1}{\sqrt{2}}[\alpha(1)\beta(2) + \beta(1)\alpha(2)]\]is also symmetric as exchanging particles keeps the combination intact.
• The function \[\frac{1}{\sqrt{2}}[\alpha(1)\beta(2) - \beta(1)\alpha(2)]\]is antisymmetric, where a swap flips the sign.

Spin symmetry is crucial as it lays the foundation for determining whether the spatial wave function needs to be symmetric or antisymmetric to follow the Pauli exclusion principle.

Spatial Wave Function

The spatial wave function describes the spatial part of a quantum system, constituting the probability amplitude for the positions of the particles. For a two-electron system, spatial symmetry is dictated by the interchange of the spatial coordinates of the electrons.

Two-electron spatial wave functions can be:

• Symmetric: \[[\psi(1) \phi(2) + \psi(2) \phi(1)]\]This expression remains unchanged under exchange of spatial coordinates of electrons, indicating symmetry.
• Antisymmetric: \[[\psi(1) \phi(2) - \psi(2) \phi(1)]\]This changes sign under electron coordinate exchange, portraying antisymmetry.

The combination of the spatial wave function with the spin function ensures the overall wave function complies with the exclusion principle. Therefore, it must be either symmetrical in spin and antisymmetrical in spatial parts or vice versa.

Singlet and Triplet States

In quantum systems, particularly in two-electron systems, the terms singlet and triplet describe particular states with different total spin characteristics. These terms arise from combining the two electron spins in distinct ways:

• Singlet State:
  This state has a total spin (\[S=0\]), leading to only one configuration, from its name "singlet." It is formed with an antisymmetric spin part;\[\frac{1}{\sqrt{2}}[\alpha(1)\beta(2) - \beta(1)\alpha(2)]\], which pairs with symmetric spatial parts to satisfy quantum requirements.
• Triplet State:
  This state features total spin \[S=1\], resulting in three possible configurations, corresponding to projections of \[M_s = 1, 0, -1\]. The spin configurations are symmetric forms such as\[\alpha(1) \alpha(2), \beta(1) \beta(2), \text{and} \frac{1}{\sqrt{2}}[\alpha(1)\beta(2) + \beta(1)\alpha(2)]\]. The triplet states are associated with an antisymmetrical spatial configuration.

Both states play a crucial role in distinguishing the different energy levels and degeneracies of quantum systems, such as in the helium atom's excited state.

Total Spin Projection

The total spin projection, denoted as \[M_s\], pertains to the sum of the spin quantum numbers of all electrons in a system, giving the component of total spin along a specified direction (usually the z-axis). For a two-electron system, the various combinations of spins yield distinct projections.

Consider these projections for both singlet and triplet states:

• Singlet State:
  With one configuration,\[M_s = 0\], as it's the only viable sum from \[S=0\].The singlet contributes a solitary degenerate level.
• Triplet States:
  For \[S=1\], the three configurations render projections of \[M_s = 1, 0, -1\].Each corresponds to different angular momentum contributions, contributing to a triply degenerate energy level.

In systems like helium's first excited state, these projections illustrate how different configurations lead to varying energy levels, bounding electron states into distinct spectral lines.