Spin-Atom Wavefunctions

The description of atoms can be formulated to a very good approximation under the assumption that the total Hamiltonian depends only on spatial coordinates (and derivatives with respect to spatial coordinates), but not on spin variables. We can, therefore, separate the stationary-state wave function according to a product of spatial and spin wavefunctions.

Example 1: The spin-atom wavefunction of the hydrogen atom can be approximated as follows,

$$\psi_{el}= \psi(x, y, z) g(m_s),$$

where $g(m_s)=\alpha,\beta$, when $m_S = 1/2,-1/2$, respectively. Since the Hamiltonian operator is assumed to be independent of spin variables, it does not affect the spin function, and the eigenvalues of the system are the same as the energies found with a wave function that did not involve spin coordinates. Mathematically,

$$\hat{H}[ \psi(x, y, z) g(m_s)] = g(m_s) \hat{H} \psi(x,y,z) = E g(m_s) \psi(x,y,z).$$

The only consequence of modeling the hydrogen atom according to a spin-atom wavefunction is that the degeneracy of the energy levels is increased.

Example 2: The ground electronic state energy of the helium atom has been modeled according to the zeroth-order wave function 1S(1) 1S(2). In order to take spin into account we must multiply such spatial wavefunction by a spin eigenfunction. Since each electron has two possible spin states, there are in principle four possible spin functions:

$$\alpha (1) \alpha(2), \qquad \alpha (1) \beta(2), \qquad \beta (1) \alpha(2),$$   and$$\qquad \beta (1) \beta(2).$$

Functions $\alpha (1) \beta(2)$, and $\beta (1) \alpha(2)$, however, are not invariant under an electron permutation (i.e., these functions make a distinction between electron 1 and electron 2). Therefore, such functions are inadequate to describe the state of a system of indistinguishable quantum particles, such as electrons. Instead of working with functions $\alpha (1) \beta(2)$ and $\beta (1) \alpha(2)$, it is necessary to construct linear combinations of such functions, e.g.,

$$\frac{1}{\sqrt{2}} \left[ \alpha (1) \beta(2) \pm \beta (1) \alpha(2) \right] ,$$

with correct exchange properties associated with indistinguishable particles,

$$\hat{P}_{12} \psi_{(1, 2)} = \pm \psi_{(2,1)}.$$

The two linear combinations, together with functions $\alpha (1) \alpha(2)$ and $\beta (1) \beta(2)$, form the basis of four normalized two-electron spin eigenfunctions of the helium atom.