Quantum Mechanics Term 1 Problems Class 3

Dr A. Donos

(Michaelmas 2021)

1) Consider a 2-D Hilbert space with orthonormal basis $\{\ket{1},\ket{2}\}$ and define an operator

\displaystyle\hat{A}=2\ket{1}\bra{1}+3\ket{2}\bra{2}+i\sqrt{2}\ket{1}\bra{2}+% \alpha\ket{2}\bra{1}

a) Find $\alpha$ so that $\hat{A}$ is an observable
b) What are the possible values that $\hat{A}$ can yield?
c) Find $\braket{\hat{A}}_{\psi},\ket{\psi}=\frac{1}{\sqrt{3}}(i\ket{1}+\sqrt{2}\ket{2})$
d) What does this tell you about the state $\ket{\psi}$ ?

Answers:
a) As indirect as it is, the expression given for $\hat{A}$ is immediately translatable to its matrix representation.

\displaystyle A=\left(\begin{array}[]{ c c}2&i\sqrt{2}\\ \alpha&3\end{array}\right)

For $\hat{A}$ to be an observable it has to be Hermitian. Hence, $\alpha=(i\sqrt{2})^{*}=-i\sqrt{2}$ .

b) The possible measurements of $\hat{A}$ are its eigenvalues

\displaystyle det(A-wI)

\displaystyle=(2-w)(3-w)-2=(w-4)(w-1)

Hence the possible measurements are 1 or 4.

c) We can do this in two ways. The first way is the formula where we evaluate the inner product $\braket{\psi}{\hat{A}}{\psi}$ as usual. Alternatively, we can evaluate this inner product using matrices.

	$\displaystyle\ket{\psi}$	$\displaystyle=\frac{1}{\sqrt{3}}\left(\begin{array}[]{c}i\\ \sqrt{2}\end{array}\right)$
	$\displaystyle\Rightarrow\bra{\psi}$	$\displaystyle=\frac{1}{\sqrt{3}}\begin{pmatrix}-i&\sqrt{2}\end{pmatrix}$
	$\displaystyle\Rightarrow\braket{A}_{\psi}$	$\displaystyle=\frac{1}{3}\begin{pmatrix}-i&\sqrt{2}\end{pmatrix}\left(\begin{% array}[]{c c}2&i\sqrt{2}\\ -i\sqrt{2}&3\end{array}\right)\left(\begin{array}[]{c}i\\ \sqrt{2}\end{array}\right)$
		$\displaystyle=4$

d) An interesting to note is that the expectation is the value of an eigenvalues. This suggests that $\ket{\psi}$ is an eigenvector with eigenvalue 4.

	$\displaystyle\braket{\hat{A}}_{\psi}$	$\displaystyle=p_{1}(1)+(1-p_{1})(4)=-3p_{1}+4$
	$\displaystyle\Rightarrow p_{1}$	$\displaystyle=0$
	$\displaystyle\Rightarrow p_{4}$	$\displaystyle=1$
	$\displaystyle p_{4}$	$\displaystyle=\braket{\psi}{\hat{P}_{A=4}}{\psi}$
	$\displaystyle\Rightarrow\hat{P}_{A=4}$	$\displaystyle=\ket{A=4}\bra{A=4}=\hat{I}$
	$\displaystyle\Rightarrow\hat{P}_{A=4}\ket{\psi}$	$\displaystyle=\ket{\psi}$
	$\displaystyle\Rightarrow\ket{\psi}$	$\displaystyle\in V_{A=4}$

2) Consider a unitary operator $\hat{U}$ with non-degenerate spectrum. Determine whether the following statements are true or false.
a) $\hat{U}^{k}$ is Unitary for any $k$ positive integer.
b) $\det(U)$ is a complex number of unit modulus.
c) The eigenvalues of $\hat{U}$ are complex numbers of unit modulus.
d) The eigenvectors of $\hat{U}$ are mutually orthogonal.

Answer:
a)

\displaystyle(\hat{U}^{k})^{+}\hat{U}^{k}=\hat{U}^{+}\dots\hat{U}^{+}\hat{U}% \dots\hat{U}=1

	$\displaystyle\det(U^{+}U)$	$\displaystyle=1$
	$\displaystyle\Leftrightarrow\det(U^{+})\det(U)$	$\displaystyle=1$
	$\displaystyle\Leftrightarrow\|det(U)\|^{2}$	$\displaystyle=1$

c) & d) Consider 2 eigenvectors

	$\displaystyle\hat{U}\ket{U_{1}}$	$\displaystyle=U_{1}\ket{U_{1}}$
	$\displaystyle\hat{U}\ket{U_{2}}$	$\displaystyle=U_{2}\ket{U_{2}}$
	$\displaystyle\Rightarrow\bra{U_{2}}\hat{U}^{+}$	$\displaystyle=\bra{U_{2}}U_{2}^{*}$
	$\displaystyle\braket{U_{2}}{U_{1}}$	$\displaystyle=\braket{U_{2}}{\hat{U}^{+}\hat{U}}{U_{1}}$
		$\displaystyle=U_{2}^{*}U_{1}\braket{U_{2}}{U_{1}}$
	$\displaystyle\Rightarrow\braket{U_{2}}{U_{1}}(1-U_{2}^{*}U_{1})$	$\displaystyle=0$

If $\ket{U_{1}}=\ket{U_{2}}$ ,

	$\displaystyle 1-\|U_{1}\|^{2}$	$\displaystyle=0$
	$\displaystyle\Rightarrow\|U_{1}\|^{2}$	$\displaystyle=1$

If $\ket{U_{1}}\neq\ket{U_{2}}$ .

	$\displaystyle U_{2}U_{2}^{*}$	$\displaystyle=1$
	$\displaystyle\Rightarrow U_{2}^{*}$	$\displaystyle=U_{2}^{-1}$
	$\displaystyle\Rightarrow\braket{U_{2}}{U_{1}}(U_{2}-U_{1})$	$\displaystyle=0$
	$\displaystyle\Rightarrow\braket{U_{2}}{U_{1}}$	$\displaystyle=0$