Collection of Solved Problems in Physics

Spin state – complex vector

Task number: 4405

• Hint – required relations

  Recall or look up the following relations:

  • the normalisation of the wavefunction

  • the matrix form of the operators corresponding to the spin projection along each axis

  • the calculation of the expectation value of an observable in a given state

  • the uncertainty of an observable in a given state

  • the form of the uncertainty relations for the spin projections along each axis

• Solution to hint – required relations

  For normalised spinors, the so-called normalisation condition must hold, \[\chi^{\dagger} \chi = 1 \, ,\] where the dagger denotes the Hermitian adjoint, i.e., the transpose of the matrix followed by complex conjugation.

  From this, the normalisation constant can be determined upon substitution and simplification.

  The matrices representing the spin-\(\frac{1}{2}\) projection along each axis are given by \[(\hat S_x, \, \hat S_y, \, \hat S_z) = \frac{\hbar}{2} \left(\begin{pmatrix} 0 &1 \\ 1& 0 \end{pmatrix},\begin{pmatrix} 0 &−i \\ i& 0 \end{pmatrix},\begin{pmatrix} 1 &0 \\ 0& −1 \end{pmatrix}\right) \, .\]

  The expectation value of the observable \(F\) in a state described by the normalised spinor \(\chi\) can be determined from \[\left \langle F \right \rangle_{\chi} = \left \langle \chi \, \Big | \, \hat F \chi \right \rangle = \chi^{\dagger} \hat F \chi \, ,\] where \(\hat F\) is the matrix of the operator associated with the observable \(F\).

  The uncertainty \(\sigma_F\) of the observable \(F\) in a state described by the spinor \(\chi\) can be obtained from \[(\sigma_F)^2 = \left \langle F^2 \right \rangle_{\chi} - \left \langle F \right \rangle_{\chi} ^2 \, .\]

  The uncertainty relations for the spin projections along the three axes in a state described by the spinor \(\chi\) are given by \[\sigma_{S_i} \, \sigma_{S_j} \geq \frac{\hbar}{2} \left | \left \langle S_k \right \rangle_{\chi} \right | \, , \qquad i, j, k = 1, 2, 3 \, .\]

• Solution a)

  We start from the normalisation condition, into which we substitute and then simplify

  \[ 1 = \chi^{\dagger}\chi = |A|^2 \begin{pmatrix}-3i & 4\end{pmatrix}\begin{pmatrix}3i\\4\end{pmatrix} = |A|^2 (9+16) = 25 |A|^2 \, . \]

  From this, we can express the magnitude of the normalisation constant

  \[ 1=25|A|^2 \, , \] \[ |A|^2 = \frac{1}{25} \, , \] \[ |A| = \frac{1}{5} \, . \]

  This determines the magnitude of the normalisation constant \(A\). In this case, the normalisation constant can take any value of the form \(\frac{1}{5}e^{i\alpha}\), where \(e^{i\alpha}\) is a complex unit. For simplicity, we choose \(A = \frac{1}{5}\).

• Solution b)

  We start from the general expression for the expectation value, into which we substitute and then simplify

  \[ \langle S_x \rangle_{\chi} = \chi^{\dagger} \hat S_x \chi = \frac{1}{5} \begin{pmatrix} -3i & 4 \end{pmatrix} \frac{\hbar}{2}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \frac{1}{5} \begin{pmatrix} 3i \\ 4 \end{pmatrix} = \frac{\hbar}{50} \begin{pmatrix} 4 & -3i \end{pmatrix} \begin{pmatrix} 3i \\ 4 \end{pmatrix} = \] \[ = \frac{\hbar}{50} (12i - 12i) = 0 \, . \]

  An analogous procedure is followed for \(S_y\) and \(S_z\)

  \[ \langle S_y \rangle_{\chi} = \chi^{\dagger} \hat S_y \chi = \frac{1}{5} \begin{pmatrix} -3i & 4 \end{pmatrix} \frac{\hbar}{2}\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} \frac{1}{5} \begin{pmatrix} 3i \\ 4 \end{pmatrix} = \frac{\hbar}{50} \begin{pmatrix} 4i & -3 \end{pmatrix} \begin{pmatrix} 3i \\ 4 \end{pmatrix} = \] \[ = \frac{\hbar}{50} (-12 - 12) = -\frac{12}{25} \hbar \, , \] \[ \langle S_z \rangle_{\chi} = \chi^{\dagger} \hat S_z \chi = \frac{1}{5} \begin{pmatrix} -3i & 4 \end{pmatrix} \frac{\hbar}{2}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \frac{1}{5} \begin{pmatrix} 3i \\ 4 \end{pmatrix} = \frac{\hbar}{50} \begin{pmatrix} -3i & -4 \end{pmatrix} \begin{pmatrix} 3i \\ 4 \end{pmatrix} = \] \[ = \frac{\hbar}{50} (9 - 16) = -\frac{7}{50} \hbar \, . \]

• Solution c)

  To calculate the uncertainty, in addition to the expectation value (see the previous section), we also need to determine the expectation value of the square of each spin projection. Noting that the square of the spin projection along each axis \(x, \, y, \, z\) is \(\frac{\hbar^2}{4} \hat {\mathbb E}\), where \( \hat {\mathbb E}\) is the \(2×2\) identity matrix, it follows that the expectation value of the square of the spin projection along any of the axis \(x, \, y, \, z\) is precisely \(\frac{\hbar^2}{4} \) in any state. A detailed calculation of this fact can be found in task Expectation values of the spin components in a general state, Solution a).

  We now simply substitute into the expression for the square of the uncertainty and simplify

  \[ \left ( \sigma_{S_x} \right )^2 = \left \langle S_x^2 \right \rangle_{\chi} - \left \langle S_x \right \rangle_{\chi} ^2 = \frac{\hbar^2}{4} - 0 = \frac{\hbar^2}{4} \, . \]

  To obtain the uncertainty, we now simply take the square root of the result

  \[ \delta S_x = \frac{\hbar}{2} \, . \]

  For \(S_y\) and \(S_z\), the procedure is entirely analogous

  \[ \left ( \delta S_y \right )^2 = \left \langle S_y^2 \right \rangle_{\chi} - \left \langle S_y \right \rangle_{\chi} ^2 = \frac{\hbar^2}{4} - (-\frac{12}{25} \hbar)^2 = \frac{49}{2500} \hbar^2 \, , \] \[ \delta S_y = \frac{7}{50} \hbar \, , \] \[ \left ( \delta S_z \right )^2 = \left \langle S_z^2 \right \rangle_{\chi} - \left \langle S_z \right \rangle_{\chi} ^2 = \frac{\hbar^2}{4} - (-\frac{7}{50} \hbar)^2 = \frac{144}{625} \hbar^2 \, , \] \[ \delta S_z = \frac{12}{25} \hbar \, . \]

• Solution d)

  Given the form of the uncertainty relations, we will verify their validity only for three distinct pairs of spin projections. We start from the general relation, substitute, simplify, and then determine whether the inequality holds:

  For the pair \(S_x\) and \(S_y\), the left-hand side is

  \[ \sigma_{S_x} \, \sigma_{S_y} = \frac{\hbar}{2} \frac{7 \hbar}{50} = \frac{7}{100} \hbar^2 \]

  and the right-hand side is

  \[ \frac{\hbar}{2} \, \left | \left \langle S_z \right \rangle_{\chi} \right | = \frac{\hbar}{2} \, \left |-\frac{7}{50} \hbar \right | = \frac{7}{100} \hbar^2 \, . \]

  It is thus clear that \(\sigma_{S_x} \, \sigma_{S_y} \geq \frac{\hbar}{2} \left | \left \langle S_z \right \rangle_{\chi} \right |\) is satisfied.

  An entirely analogous procedure is followed for the pair \(S_x\) and \(S_z\)

  \[ \sigma_{S_x} \, \sigma_{S_z} = \frac{\hbar}{2} \frac{12 \hbar}{25} = \frac{6}{25} \hbar^2 \, , \] \[ \frac{\hbar}{2} \, \left | \left \langle S_y \right \rangle_{\chi} \right | = \frac{\hbar}{2} \, \left |-\frac{12}{25} \hbar \right | = \frac{6}{25} \hbar^2 \, . \]

  It is thus clear that \(\delta S_x \, \delta S_z \geq \frac{\hbar}{2} \left | \left \langle S_y \right \rangle_{\chi} \right |\) is satisfied.

  For the pair \(S_y\) and \(S_z\), we obtain

  \[ \sigma_{S_y} \, \sigma_{S_z} = \frac{7 \hbar}{50} \frac{12 \hbar}{25} = \frac{42}{625} \hbar^2 \, , \] \[ \frac{\hbar}{2} \, \left | \left \langle S_x \right \rangle_{\chi} \right | = \frac{\hbar}{2} \cdot 0 = 0 \, . \]

  It is thus clear that \(\delta S_y \, \delta S_z \geq \frac{\hbar}{2} \left | \left \langle S_x \right \rangle_{\chi} \right |\) is also satisfied.

  Note: This case is particularly interesting, as it shows that even for non-commuting operators, the expectation value of their commutator can be zero in a specific state.

• Answer

  a) The normalisation constant \(A\) has a magnitude of \(\frac{1}{5}\).

  b) The expectation values of the spin projections along the \(x, \, y, \, z\) axes in the given state are \[ \left \langle S_x \right \rangle_{\chi} = 0 \, , \] \[ \left \langle S_y \right \rangle_{\chi} = - \frac{12}{25} \hbar \, , \] \[ \left \langle S_z \right \rangle_{\chi} = - \frac{7}{50} \hbar \, . \]

  c) The uncertainties of the spin projections along the \(x, \, y, \, z\) axes in the given state are: \[ \sigma_{S_x} = \frac{\hbar}{2} \, , \] \[ \sigma_{S_y} = \frac{7}{50} \hbar \, , \] \[ \sigma_{S_z} = \frac{12}{25} \hbar \, . \]

  d) We have verified the validity of the uncertainty relations in the given state for all pairs of spin projections along the individual axes.