Question

An electron is in the spin state

$\mathcal{x}\mathbf{=}\mathbf{A}\left(\begin{array}{c}1-2i\\ 2\end{array}\right)$

(a) Determine the constant by normalizing $\mathcal{x}$.

(b) If you measured ${\mathit{S}}_{\mathbf{z}}$on this electron, what values could you get, and what is the probability of each? What is the expectation value of ${\mathbf{S}}_{\mathbf{z}}$?

(c) If you measured ${\mathbf{S}}_{\mathbf{x}}$on this electron, what values could you get, and what is the probability of each? What is the expectation value of ${\mathbf{S}}_{\mathbf{x}}$?

(d) If you measured ${\mathbf{S}}_{\mathbf{y}}$on this electron, what values could you get, and what is the probability of each? What is the expectation value of${\mathbf{S}}_{\mathbf{y}}$?

Short answer

(a) The constant by normalizing $\mathcal{x}\mathrm{is}\mathrm{A}=\frac{1}{3}$

(b) The value $\frac{\sout{h}}{2}$and $-\frac{\sout{h}}{2}$has a probability of $\frac{5}{9}$and $\frac{4}{9}$respectively. The expectation value of

(c) The value $\frac{\sout{h}}{2}$and $\frac{-\sout{h}}{2}$has a probability of $\frac{13}{18}$and $\frac{5}{18}$. The expectation value of

(d) The value $\frac{\sout{h}}{2}$and $\frac{-\sout{h}}{2}$has a probability of $\frac{17}{18}$and $\frac{1}{18}$. The expectation value of

Step-by-step solution

Determine the eigen functions

The Hamiltonian operator's eigen functions k are stationary states of the quantum mechanical system, each with its own energy . They reflect the system's allowed energy states and may be confined by boundary requirements.

Determine the constant A by normalizing  x

(a)

We have an electron in a spin state given by:

$\mathcal{x}=A\left(\begin{array}{c}1-2i\\ 2\end{array}\right)$ …… (1)

To find the constant we set:

$$\begin{gathered} {\left|\mathcal{x}\right|}^2=1 \\ {\left|A\right|}^2\left(5+4\right)=1 \\ 9{\left|A\right|}^2=1 \\ A=\frac{1}{3} \end{gathered}$$

Therefore, the constant by normalizing $\mathcal{x}isA=\frac{1}{3}$.

Determine the expectation value of  Sz

(b)

To find $S_z$we need to write $\mathcal{x}$as a linear combination of the eigenspinors of $S_z$as:

$\mathcal{x}=a\left(\begin{array}{c}1\\ 0\end{array}\right)+b\left(\begin{array}{c}0\\ 1\end{array}\right)$

Compare with equation (1) we get:

$\mathcal{x}=\frac{1-2i}{3}\left(\begin{array}{c}1\\ 0\end{array}\right)+\frac{2}{3}\left(\begin{array}{c}0\\ 1\end{array}\right)$

The possible values of localid="1656332066764" $S_{z}are\sout{h}/2and-\sout{h}/2$with probabilities of ${\left|\left(1-2i\right)/3\right|}^2=5/9$and 4/9 that is: $\frac{\sout{h}}{2}$has a probability of $\frac{5}{9}$, and $\frac{\sout{h}}{2}$ with probability $\frac{4}{9}$.

The expectation value of $S_z$is:

We know the following values,

$$\begin{gathered} {\mathcal{x}}^T=\frac{1}{9}\left(1+2i2\right) \\ {S}_z=\frac{\sout{h}}{2}\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right) \\ \mathcal{x}=\left(\begin{array}{c}1-2i\\ 2\end{array}\right) \end{gathered}$$

The expectation value will be,

Therefore, the expectation value of.

Determine the expectation value of Sx

(c)

The operator S_x is given by,

$S_x=\frac{\sout{h}}{2}\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

Find the eigenvalues of this operator as:

$$\begin{gathered} \left|\begin{array}{cc}-\lambda & -\frac{\sout{h}}{2}\\ \frac{\sout{h}}{2}& -\lambda \end{array}\right|=0 \\ {\lambda }^2-\frac{{\sout{h}}^2}{4}=0 \\ \lambda =\pm \frac{\sout{h}}{2} \end{gathered}$$

Solving the equation to find the eigenspinors:

$\frac{\sout{h}}{2}\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)\left(\begin{array}{c}\propto \\ \beta \end{array}\right)=\pm \frac{\sout{h}}{2}\left(\begin{array}{c}\propto \\ \beta \end{array}\right)$

The eigenspinors are,

$$\begin{gathered} {\mathcal{x}}_{+}^{\left(x\right)}=\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ 1\end{array}\right) \\ {\mathcal{x}}_{+}^{\left(x\right)}=\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ -1\end{array}\right) \end{gathered}$$ …… (2)

Write the eigenspinors of $S_x$as:

$\mathcal{x}=\frac{\sqrt{2}}{3}\left[a\left(\begin{array}{c}1\\ 1\end{array}\right)+b\left(\begin{array}{c}1\\ -1\end{array}\right)\right]$

To find the constant a and b . We set this equation equal to equation ( I )as:

$$\begin{gathered} a+b=1-2 \\ a-b=2 \\ a=\frac{3}{2}-i \\ b=-\frac{1}{2}-i \end{gathered}$$

The eigenspinor will be:

$\mathcal{x}=\frac{\sqrt{2}}{3}\left[\left(\frac{3}{2}-i\right)\left(\begin{array}{c}1\\ 1\end{array}\right)+\left(-\frac{1}{2}-i\right)\left(\begin{array}{c}1\\ -1\end{array}\right)\right]$

According to this equation the possible values are $\sout{h}/2$with probability ${\left|\left(\sqrt{2}/3\right)\left(3/2-i\right)\right|}^2=13/18$and $-\sout{h}/2$with probability 5/18.

That is: $\frac{\sout{h}}{2}$, with probability $\frac{13}{18}$; $-\frac{\sout{h}}{2}$, with probability$\frac{5}{18}$

The expectation value of $S_x$is:

We know the following values,

$$\begin{gathered} {\mathcal{x}}^T=\frac{1}{9}\left(1+2i2\right) \\ {S}_x=\frac{\sout{h}}{2}\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right) \\ \mathcal{x}=\left(\begin{array}{c}1-2i\\ 2\end{array}\right) \end{gathered}$$

The expectation value will be,

Therefore, the expectation value of

Determine the expectation value of Sy

(d)

In this part we follow the same method in part to find the eigenvalues and their probabilities,

$\mathcal{x}=\frac{\sqrt{2}}{3}\left[\left(\frac{1}{2}-2\mathrm{i}\right)\left(\begin{array}{c}1\\ \mathrm{i}\end{array}\right)+\frac{1}{2}\left(\begin{array}{c}1\\ -\mathrm{i}\end{array}\right)\right]$

The possible values are $\sout{h}/2$with probability ${\left|\sqrt{2}/3\left(1/2-2\mathrm{i}\right)\right|}^2=17/18$and $-\sout{h}/2$with probability 1/18

That is: $\frac{\sout{h}}{2}$, with probability $\frac{17}{18}$; $-\frac{\sout{h}}{2}$, with probability$\frac{1}{18}$.

The expectation value of$S_{y}is:$

Therefore, the expectation value of.