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Physics LibreTexts

5.3: Wavefunction of Spin One-Half Particle

The state of a spin one-half particle is represented as a vector in ket space. Let us suppose that this space is spanned by the basis kets $ \vert x', y', z', \pm\rangle$ . Here, $ \vert x', y', z', \pm\rangle$ denotes a simultaneous eigenstate of the position operators $ x$ , $ y$ , $ z$ , and the spin operator $ S_z$ , corresponding to the eigenvalues $ x'$ , $ y'$ , $ z'$ , and $ \pm \hbar/2$ , respectively. The basis kets are assumed to satisfy the completeness relation

 

$\displaystyle \int\!\int\!\int dx'dy'dz' \left(\, \vert x',y',z',+\rangle\langle x', y', z',+\vert+\vert x',y',z',-\rangle\langle x', y', z',-\vert \,\right)= 1.$ (431)

 

 

It is helpful to think of the ket $ \vert x', y', z', +\rangle$ as the product of two kets--a position space ket $ \vert x', y', z'\rangle$ , and a spin space ket $ \vert+\rangle$ . We assume that such a product obeys the commutative and distributive axioms of multiplication:

 

$\displaystyle \vert x', y', z'\rangle \vert+\rangle$ $\displaystyle = \vert+\rangle \vert x', y', z'\rangle,$ (432)
$\displaystyle \left(c'\, \vert x', y', z'\rangle + c''\,\vert x'', y'', z''\rangle\right)\, \vert+\rangle$ $\displaystyle = c'\, \vert x', y', z'\rangle \vert+\rangle+ c'' \,\vert x'', y'', z''\rangle \vert+\rangle,$ (433)
$\displaystyle \vert x', y', z'\rangle\left(c_+ \,\vert+\rangle + c_-\, \vert-\rangle\right)$ $\displaystyle = c_+ \, \vert x', y', z'\rangle\vert+\rangle+ c_-\,\vert x', y', z'\rangle\vert-\rangle,$ (434)

 

 

where the $ c$ 's are numbers. We can give meaning to any position space operator (such as $ L_z$ ) acting on the product $ \vert x', y', z'\rangle
\vert+\rangle$ by assuming that it operates only on the $ \vert x', y', z'\rangle$ factor, and commutes with the $ \vert+\rangle$ factor. Similarly, we can give a meaning to any spin operator (such as $ S_z$ ) acting on $ \vert x', y', z'\rangle
\vert+\rangle$ by assuming that it operates only on $ \vert+\rangle$ , and commutes with $ \vert x', y', z'\rangle$ . This implies that every position space operator commutes with every spin operator. In this manner, we can give meaning to the equation

 

$\displaystyle \vert x', y', z', \pm\rangle = \vert x', y', z'\rangle\vert \pm\rangle = \vert \pm\rangle \vert x', y', z'\rangle.$ (435)

 

 

The multiplication in the above equation is of a quite different type to any that we have encountered previously. The ket vectors $ \vert x', y', z'\rangle$ and $ \vert\pm \rangle$ lie in two completely separate vector spaces, and their product $ \vert x',y', z'\rangle\vert\pm\rangle$ lies in a third vector space. In mathematics, the latter space is termed the product space of the former spaces, which are termed factor spaces. The number of dimensions of a product space is equal to the product of the number of dimensions of each of the factor spaces. A general ket of the product space is not of the form (435), but is instead a sum or integral of kets of this form.

A general state $ A$ of a spin one-half particle is represented as a ket $ \vert\vert A\rangle\rangle$ in the product of the spin and position spaces. This state can be completely specified by two wavefunctions:

 

$\displaystyle \psi_+(x', y', z')$ $\displaystyle = \langle x', y', z' \vert\langle +\vert\vert A\rangle\rangle,$ (436)
$\displaystyle \psi_-(x', y', z')$ $\displaystyle = \langle x', y', z' \vert\langle -\vert\vert A\rangle\rangle.$ (437)

 

 

The probability of observing the particle in the region $ x'$ to $ x'+dx'$ , $ y'$ to $ y'+dy'$ , and $ z'$ to $ z'+dz'$ , with $ s_z = +1/2$ is $ \vert\psi_+ (x', y', z')\vert^{\,2}\,dx' dy' dz'$ . Likewise, the probability of observing the particle in the region $ x'$ to $ x'+dx'$ , $ y'$ to $ y'+dy'$ , and $ z'$ to $ z'+dz'$ , with $ s_z = -1/2$ is $ \vert\psi_- (x', y', z')\vert^{\,2}\,dx' dy' dz'$ . The normalization condition for the wavefunctions is

 

$\displaystyle \int\!\int\!\int dx'dy'dz' \left(\vert\psi_+\vert^{\,2} + \vert\psi_-\vert^{\,2}\right)= 1.$ (438)

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