Chapter 4: Q27P (page 177)

An electron is in the spin state
$χ = A (\begin{matrix} 3 i \\ 4 \end{matrix})$
(a) Determine the normalization constant .
(b) Find the expectation values of $S_{x}$ , $S_{y}$ , and $S_{z}$ .
(c) Find the "uncertainties" , $σ_{S_{x}}$ , $σ_{S_{y}}$ and $σ_{S_{z}}$ . (Note: These sigmas are standard deviations, not Pauli matrices!)
(d) Confirm that your results are consistent with all three uncertainty principles (Equation 4.100 and its cyclic permutations - only with in place ofL, of course).

Short Answer

Expert verified

The normalization constant is $A = \frac{1}{5}$
Theexpectation values of,, andis, and.
Theexpectation values of,, andis, andrespectively
The values of all three uncertainty principles , , and is , and .

Step by step solution

Significance of electron in spin state

The angular momentum of the electrons is used to describe the spin of the electrons. It only spins in two directions, that is up and down direction.

(a) Determine the normalization constant.

Determine the normalization constant by using $⟨ χ ∣ χ ⟩ = 1$

$\begin{array}{rcl} A^{2} (\begin{matrix} - 3 i & 4 \end{matrix}) (\begin{matrix} 3 i \\ 4 \end{matrix}) & = & 1 \\ A^{2} (9 + 16) & = & 1 \\ A^{2} & = & \frac{1}{25} \\ A & = & \frac{1}{5} \end{array}$

Thus, the normalization constant is $A = \frac{1}{5}$

(b) Find the expectation values of Sx, Sy, and Sz

Determinethe expectation values of $S_{x}$ .

$\begin{array}{rcl} <S_{x}> & = & <χ |S_{x}| χ> \\ = & \frac{ℏ}{50} (\begin{matrix} - 3 i & 4 \end{matrix}) (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}) (\begin{matrix} 3 i \\ 4 \end{matrix}) \\ = & \frac{ℏ}{50} (\begin{matrix} - 3 i & 4 \end{matrix}) (\begin{matrix} 4 \\ 3 i \end{matrix}) \\ = & \frac{ℏ}{50} (12 i - 12 i) \end{array}$

Determine theexpectation values of $S_{y}$ .

$\begin{array}{rcl} <S_{y}> & = & <χ |S_{y}| χ> \\ = & \frac{ℏ}{50} (\begin{matrix} - 3 i & 4 \end{matrix}) (\begin{matrix} 0 & - i \\ i & 0 \end{matrix}) (\begin{matrix} 3 i \\ 4 \end{matrix}) \\ = & \frac{ℏ}{50} (\begin{matrix} - 3 i & 4 \end{matrix}) (\begin{matrix} - 4 i \\ 3 \end{matrix}) \\ = & \frac{ℏ}{50} (- 12 - 12) \\ = & \frac{- 24 ℏ}{50} \end{array}$

Determine the expectation values of $S_{z}$

$\begin{array}{rcl} <S_{z}> & = & <χ |S_{z}| χ> \\ = & \frac{ℏ}{50} (\begin{matrix} - 3 i & 4 \end{matrix}) (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}) (\begin{matrix} 3 i \\ 4 \end{matrix}) \\ = & \frac{ℏ}{50} (\begin{matrix} - 3 i & 4 \end{matrix}) (\begin{matrix} 3 i \\ - 4 \end{matrix}) \\ = & \frac{ℏ}{50} (\begin{matrix} 9 - 16 \end{matrix}) \\ = & \frac{- 7 ℏ}{50} \end{array}$

Thus, the expectation values of $S_{x}$ , $S_{y}$ , and $S_{z}$ is 0, $\frac{- 24 ℏ}{50}$ and $\frac{- 7 ℏ}{50}$

(c) Determination of the value of σSx,σSy , and σSz

Determine the expectation values of $σ_{S_{x}}$

$\begin{array}{rcl} d σ_{S_{x}}^{2} & = & <S_{x}^{2}> - {<S_{x}>}^{2} \\ σ_{S_{x}}^{2} & = & \frac{ℏ^{2}}{4} - 0 \\ σ_{S_{x}} & = & \frac{ℏ}{2} \end{array}$

Determine the expectation values of $σ_{S_{y}}$ .

$σ_{S_{y}}^{2} = <S_{y}^{2}> - {<S_{y}>}^{2} σ_{S_{y}}^{2} = \frac{ℏ^{2}}{4} - {(\frac{24}{50})}^{2} ℏ^{2} σ_{S_{y}}^{2} = \frac{49}{2500} ℏ^{2} σ_{S_{y}} = \frac{7}{50} ℏ$

Determine the expectation values of $σ_{S_{z}}$

$σ_{S_{z}}^{2} = <S_{z}^{2}> - {<S_{z}>}^{2} σ_{S_{z}}^{2} = \frac{ℏ^{2}}{4} - {(\frac{7}{50})}^{2} ℏ^{2} σ_{S_{z}}^{2} = \frac{576}{2500} ℏ^{2} σ_{S_{z}} = \frac{12}{25} ℏ$

Thus, the expectation values of $σ_{S_{x}}$ , $σ_{S_{y}}$ , and is $\frac{ℏ}{2}$ , $\frac{7}{50} ℏ$ and $\frac{12}{25} ℏ$ respectively

(d) Confirmation of the consistency of the results of all three uncertainty principles

Determine the value of $σ_{S_{x}} σ_{S_{y}}$ to confirm the result.

$\begin{array}{rcl} σ_{S_{x}} σ_{S_{y}} & = & \frac{ℏ}{2} \cdot \frac{7 ℏ}{50} \\ = & \frac{ℏ}{2} |<S_{z}>| \\ = & \frac{ℏ}{2} \cdot \frac{7 ℏ}{5} \end{array}$

Determine the value of $σ_{S_{y}} σ_{S_{z}}$ to confirm the result.

$\begin{array}{rcl} σ_{S_{y}} σ_{S_{z}} & = & \frac{7 ℏ}{50} \cdot \frac{12}{25} ℏ > \frac{ℏ}{2} |<S_{x}>| \\ = & 0 \end{array}$

Determine the value of $σ_{S_{z}} σ_{S_{x}}$ to confirm the result.

$\begin{array}{rcl} σ_{S_{z}} σ_{S_{x}} & = & \frac{12}{25} ℏ \cdot \frac{ℏ}{2} \\ = & \frac{ℏ}{2} |<S_{y}>| \\ = & \frac{ℏ}{2} \cdot \frac{12}{25} ℏ \end{array}$

In general, $σ_{S_{i}} σ_{S_{j}} ⩾ \frac{ℏ}{2} |<S_{k}>|$ and cyclic permutation.

Thus, the valuesof all three uncertainty principles $σ_{S_{x}} σ_{S_{y}}$ , $σ_{S_{y}} σ_{S_{z}}$ , and $σ_{S_{z}} σ_{S_{x}}$ is $\frac{ℏ}{2} \cdot \frac{7 ℏ}{5}$ , 0 and $\frac{ℏ}{2} \cdot \frac{12}{25} ℏ$ .

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Short Answer

Step by step solution

Significance of electron in spin state

(a) Determine the normalization constant.

(b) Find the expectation values of Sx, Sy, and Sz

(c) Determination of the value of σSx,σSy , and σSz

(d) Confirmation of the consistency of the results of all three uncertainty principles

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