Problem set 39

1. A state of a system with spherically symmetric potential has zero uncertainty in simultaneous measurement of operator $L_x$ and $L_y$. Which of the following statement is true?
1. The state must be $l=0$ state
2. Such a state can never exist as operators $L_x$ and $L_y$ do not commute
3. The state has $l=1$ with $m=0$
4. The state cannot be an eigenstate of $L^2$ operator
$$\begin{align*} L_z\psi_{nlm}&=m\hbar\psi_{nlm} \text{ and } L^2\psi_{nlm}\\ &=l(l+1)\hbar^2\psi_{nlm} \end{align*}$$ $$\left < L_z \right > =m\hbar, \left < L_z^2 \right > =m^2\hbar^2$$ $$\text{ and } \left < L^2 \right > =l(l+1)\hbar^2$$ The state $\psi_{nlm}$ is symmetric in $x$ and $y$, so $$\begin{align*} \left < L_x^2 \right >&=\left < L_y^2 \right > \\ &=\frac{\left(\left < L^2 \right >-\left < L_z^2 \right >\right)}{2}\\ &=\frac{\left(l(l+1)-m^2\right)\hbar^2}{2} \end{align*}$$ It can be verified that $\left < L_x \right >=\left < L_y \right >=0$. Using, $\Delta L_i=\sqrt{\left < L_i^2 \right >-\left < L_i \right >^2}$, we have $$\Delta L_x=\Delta L_y=\sqrt{\frac{\left(l(l+1)-m^2\right)\hbar^2}{2}}$$ $$\Delta L_x\Delta L_y=\frac{\left(l(l+1)-m^2\right)\hbar^2}{2}$$ From this relation one can see that uncertainty relation will be zero when $l=0$ and henc, $m=0$.
Hence, answer is (A)
2. The wave function for identical fermions is antisymmetric under particle exchange. Which of the following is a consequence of this property?
1. Heisenberg's uncertainty principle
2. Bohr correspondence principle
3. Bose-Einstein condensation
4. Pauli exclusion principle
(D) Pauli exclusion principle
3. The spin part of two electron wave function is described as a triplet state. The space part of the wave function is given by ($\psi_1$ and $\psi_2$ are two different functions):
1. $\psi_1(r_1)\psi_2(r_2)$
2. $\psi_1(r_1)\psi_2(r_2)-\psi_2(r_1)\psi_1(r_2)$
3. $\psi_1(r_1)\psi_2(r_2)+\psi_2(r_1)\psi_1(r_2)$
4. $\psi_1(r_1)\psi_1(r_2)+\psi_2(r_1)\psi_1(r_2)$
Triplet state means $S=s_1+s_2=1$ and $M_s=-1,0,+1$. For $M_s=-1$ we must have $m_{s1}=-\frac{1}{2}$ and $m_{s2}=-\frac{1}{2}$
For $M_s=0$ we must have either $m_{s1}=\frac{1}{2}$ and $m_{s2}=-\frac{1}{2}$ or $m_{s1}=-\frac{1}{2}$ and $m_{s2}=\frac{1}{2}$
For $M_s=+1$ we must have $m_{s1}=+\frac{1}{2}$ and $m_{s2}=+\frac{1}{2}$
Hence, we have three possible spin eigenfunctions $$\phi\left(+\frac{1}{2},+\frac{1}{2}\right)$$ $$\frac{1}{\sqrt{2}}\left[\phi\left(+\frac{1}{2},-\frac{1}{2}\right)+\phi\left(-\frac{1}{2},+\frac{1}{2}\right)\right]$$ $$\phi\left(-\frac{1}{2},-\frac{1}{2}\right)$$ All these functions are symmetric. However, total wavefunction must be antisymmetric. Hence, space part must be antisymmetric.
Hence, answer is (B) i.e. $\psi_1(r_1)\psi_2(r_2)-\psi_2(r_1)\psi_1(r_2)$
4. A transition in which one photon is radiated by the electron in a hydrogen atom when the electron's wave function changes from $\psi_1$ to $\psi_2$ is forbidden if $\psi_1$ and $\psi_2$
1. have opposite parity
2. are both spherically symmetrical
3. are orthogonal to each other
4. are zero at the center of the atomic nucleus
Selection rules:

1.	Principal quantum number	:	$\Delta n=$ anything
2.	Orbital angular momentum	:	$\Delta l = \pm1$
3.	Magnetic quantum number	:	$\Delta m_l = 0, \pm1$
4.	Spin	:	$\Delta s = 0$
5.	Total angular momentum	:	$\Delta j = 0, \pm 1,{\scriptstyle \text{ but} j = 0 \nrightarrow j= 0}$

A. FALSE: It's not related to the selection rules.
B. TRUE: If both initial and final states have spherically symmetrical wave functions, then they have the same angular momentum. $l = 0 \rightarrow l = 0$ is forbidden as $\Delta l=0$ in this case.
C. FALSE: In any transition, eigenstates should always be mutually orthogonal.
D. FALSE: Eigenstates zero at the center $\Rightarrow l> 0$ could change, for example from 3 to 2. This is allowed.
Hence, answer is (B)
5. The puzzle of magic numbers for nuclei was resolve by :
1. introducing hard-core potential
2. introducing Yukawa potential for shell model
3. introducing tensor character to nuclear force
4. introducing spin-orbit part in the nuclear potential
(D) introducing spin-orbit part in the nuclear potential