HW 11: Hydrogen Atom and Electron Spin

Question-1¶

Probabilistic calculations with hydrogen atom wave-functions: $\psi_{n,l,m_l} = R_{n,l}(r)Y_{l,m_l}(\theta,\phi)$:

Consider a hydrogen atom which is in the following excited state (denoted as $2p$ orbital):

• How many nodes does this wave function have due to radial part and due to angular part?

• Sketch the radial and angular parts of the eigenfunctions.

• What would be the probability distribution P(r) of finding and electron at a distance (r, r+dr)?

• Prove that the wave function is normalized.

• Find the average distance for finding the electron.

• Find the most probable distance for finding the electron.

• What is the probability of finding electron within a spherical shell of one bohr raidus $a_0$ ?

Question-2¶

Superposition states of hydrogen atom.

Suppose the hydrogen atom is in a quantum superopsoition state described by the following three different eigenfunctions $\psi_{n,l,m_l}$:

• Normalize the funciton, e.g find the value for A that ensures normalization of the $\Psi$

• If we measure the values of Energy, angular momentum and its projection, what values will we obtain and with what probabilities?

• Compute the average energy, angular momentum and its projection.

• Write down the time dependent wave function $\Psi(t)$

• Will average value $\langle r \rangle$ change over time?

Question-3¶

Spin is an intrinsic property of subatomic particles manifested in having an intrinsic mangeitc moment which is not due to “rotational motion”. Spin is incorporated in wave functions by multiplication of a spin eigenfunction, often denoted via $\alpha =\psi_{+1/2}(s)\,\,\, \beta=\psi_{-1/2}(s)$
Notice that spin momentum in contrast to angular momentum has fixed value for all electrons $s=1/2$ and $(2s+1)=2$ number of posisble projections on z-axis $m_s=+1/2,-1/2$

• Write down eigenfunction-eigenvalue equation (e.g $\hat{A}\psi=a\psi$) for each operator $\hat{L},\hat{L}_z, \hat{S},\hat{S}_z$ given the following three states of hydrogen atom

  1. $\mid 2,1,1,+1/2 \rangle$

  2. $\psi_{1,0,0}\beta$

  3. $\psi_{8,2,-1}\alpha$

• State of the hydrogen atom is described by the following wavefunctions: When you do an experiment emasuring spin state, what is a probability of detecting spin being up ($m_s=+1/2$) vs down$m_s=-1/2$?

  1. $\Psi = \Big(\frac{1}{4} \Big)^{1/2} \psi_{1,0,0}\alpha+ \Big(\frac{3}{4} \Big)^{1/2}\psi_{1,0,0}\beta$

  2. $\Psi = \frac{1}{\sqrt{2}} \mid 1,1,0,+1/2 \rangle - \frac{i}{\sqrt{2}}\mid 2,1,-1,-1/2\rangle$

• Which of the following eigenfunctions are orthogonal:

  1. $\psi_{1,0,0}\alpha$ and $\psi_{1,0,0}\beta$

  2. $\psi_{2,1,-1}\alpha$ and $\psi_{2,1,0}\alpha$

  3. $\psi_{3,1,-1}\beta$ and $\psi_{3,1,-1}\beta$

Question-4¶

Consider $2p$ and $3d$ states of hydrogen atom.

• Explain why energy of hydrogen depends on quantum number n in the absence of mangietc field but depends on two quantum numbers in the presence of magnetic field.

• When placed in a magnetic field how would the degeneracy of energy levels with rescpet to $m_l$ be affected? Sketch a before after figure similiar to Fig 7.9 of the book.

• Compute energy of excitation for $2p \rightarrow 3d$ in the absence of magnetic field and compare this energy with the spacing between upper and lower energy levels of $3d$ in magnetic field $B_z=1\, tesla$. Study example 7-5 of the book.

Question-5¶

Total angular momentum J=L+S and spin-orbit coupling. [To answer this question you need to study the chapters 7.6-7.8 ]

• What is a spin-orbit coupling?

• What are term symbols?

• What are selection rules for electornic transitions?

• Why do we need to introduce new angular momentum operators $\hat{J} \,and \, \hat{J}_z$ ?

• What is the value of total angular momentum $J$ and total orbital momentum $L$ for a term: $^2 D_{5/2}$? Write down possible values of z projection for the total angular momentum $J_z$.

• Why are there 10 spectral lines observed $1s\rightarrow 2p$ transition in a weak magnetic field?

• Using the data in table 7.7 calculate the highest frequency transition between 2p and 1s state of hydrogen atom placed in magnetic field of $B_z=1\, tesla$ . Notice that compared to the last porblem this time we are accounting for the fine structure of energy levels due to spin-orbit coupling!