Spin

Spin#

What you need to know

Magnetism results from the circular motion of charged particles.
It is expected that atoms that have electrons with non-zero value of orbital angular momentum will generate mangetic moments and hence will be affected by the presence of external magnetic field.
Experiment of Stern and Herlach estblaished the existance of intrinsic magnetic moment of lectrons termed spin that are affected by magnetic field regardless of orbital angular momentum.
Contrary to the suggestive name, spin is an intrinsic magnetic momentum* which is permanently attached to a subatomic particle and has nothing to do with “spinning” or motion. Spins is a fundamental property of particles just like mass or charge.
Particles with half-integer spins, such as 1/2, 3/2, 5/2, are known as fermions, while those particles with integer spins, such as 0, 1, 2, are known as bosons. The two families of particles obey different rules. Fermions obey the Pauli exclusion principle: there cannot be two identical fermions simultaneously having the same quantum numbers, e.g, having the same position, velocity and spin direction. In contrast, bosons have no such restriction, so they may “bunch together” even if in identical states.
Normal Zeeman effect: splitting of singlet states with spin zero in the magnetic field is due to electron’s angular momentum can be understood classically.
Anomalous Zeeman effect: Is a more general case of electron spin and angular mometnum both contributing to splitting of energy levels.

Rotating charge generates mangetic moment#

electro — Fig. 35 A GIF demonstrating the existence of a magnetic field around a current carrying wire.#

A moving charge interacts with an external magnetic field.
When an electron is in a state with \(l > 0\), one can think of a circular motion of charge (of the wavefunction describing electron) around the nucleus and generate its own magnetic field.
Note that this motion is not classical but here we are just trying to obtain a wire frame model based on classical interpretation.

electro-moment — Fig. 36 Shown are magnetic moment and angular momentum generated by a charge moving on an obrit.#

Magnetic moment of electron#

According to classical mechanics a charged particle like an electron that is rotating around an axis has a magnetic moment given by:

\[{\vec{\mu} = \gamma_e\vec{L}}\]

where \(\gamma_e\) is the magnetogyric ratio of the electron expressed via fundamental constants (\(-\frac{e}{2m_e}\)) We choose the external magnetic field to lie along the \(z\)-axis and therefore it is important to consider the \(z\) component of \(\vec{\mu}\):

\[{\mu_z = -\left(\frac{e}{2m_e}\right)L_z = -\left(\frac{e\hbar}{2m_e}\right)m \equiv -\mu_B m}\]

where \(\mu_B\) is the Bohr magneton as defined above. The interaction between a magnetic moment and an external magnetic field is given by (classical expression):

\[{U = -\vec{\mu}\cdot\vec{B} = -|\vec{\mu}||\vec{B}|\cos(\alpha)}\]

Where \(\alpha\) is the angle between the two magnetic field vectors. This gives the energy for a bar magnet in presence of an external magnetic field:

Unlike quantum mechanics in classical mechanics any orientation is allowed. When the external magnetic field is oriented along \(z\)-axis the potential energy of interaction with field in classical mechanics is:

\[{U = -\mu_z B = \frac{eB}{2m_e}L_z}\]

Effect of magnetic field on atoms#

In quantum mechanics, a magnetic moment (here corresponding to a non \(s\) orbital electron) may only take specific orientations!
The \(z\)-axis is often called the quantization axis. The eigenvalues of \(\hat{L}_z\) essentially give the possible orientations of the magnetic moment with respect to the external field.
For example, consider an electron on \(2p\) orbital in a hydrogenlike atom. The electron may reside on any of \(2p_{+1}\), \(2p_0\) or \(2p_{-1}\) orbitals (degenerate without the field). For these orbitals \(L_z\) may take the following values (\(+\hbar, 0, -\hbar\)):

\[{\hat{L}_z|p_{+1}\rangle = +1\times\hbar|p_{+1}\rangle\textnormal{,\,\, }\hat{L}_z|p_0\rangle = 0\times\hbar|p_{0}\rangle\textnormal{,\,\, }\hat{L}_z|p_{-1}\rangle = -1\times\hbar|p_{-1}\rangle}\]

Mangetic field modifies the Hamiltonian#

The relative orientations with respect to the external magnetic field are shown on the left side of the figure. The total quantum mechanical Hamiltonian for a hydrogenlike atom in a magnetic field can now be written as:

\[{\hat{H} = \hat{H}_0 + \frac{eB}{2m_e}\hat{L}_z}\]

where \(\hat{H}_0\) denotes the Hamiltonian in absence of the magnetic field.
Since projection of angular momentum commutes with Hamitlonian \([\hat{L}_z, \hat{H}_0]=0\) we have same eigenfunctions for both:

\[\hat{L}_z\mid n,l, m_l\rangle = \hbar m_l \mid n,l, m_l\rangle\]

\[\hat{H}_0\mid n,l, m_l\rangle =E_n\mid n,l, m_l\rangle\]

The eigenvalues are (derivation not shown):

\[{E_{nlm} = -\frac{m_ee^4Z^2}{2(4\pi\epsilon_0)^2n^2\hbar^2} + \mu_Bm_lB}\]

where \(n = 1, 2, ...\); \(l = 0, 1, ..., n - 1\); and \(m_l = -l, ..., 0, ..., +l\). In the presence of magnetic field, the \((2l + 1)\) degenerate levels have been split (i.e., the degeneracy is lifted). This is called the orbital Zeeman effect.

The Zeeman effect#

Energy levels are affected by the external magnetic field witht the energy now depending on magnetic quantum number \(m_l\) (do not confuse with mass!)

\[ E=E_n +\hbar m_l \frac{e}{2m_e} B=E_n+m_l\beta_B B \]

Where we have introduce the Boh’r magneton \(\beta_B = \frac{\hbar e}{2 m_e}\). The energy expression predicts removal of degeneracy with respect to \(m_l\) generating splitting energy levels into \(2l+1\) lines. For instance the 2p level of hydrogen atom will split into 3 levels \(-\beta_B B, 0,+\beta_B\)

When an external magnetic field is applied, sharp spectral lines like the \(n=3\rightarrow 2\) transition of hydrogen split into multiple closely spaced lines. First observed by Pieter Zeeman, this splitting is attributed to the interaction between the magnetic field and the magnetic dipole moment associated with the orbital angular momentum. In the absence of the magnetic field, the hydrogen energies depend only upon the principal quantum number n , and the emissions occur at a single wavelength.

Disocvering Spin#

Spin as a tiny magnet#

We come to view spin of subatomic particles in a same way as we view mass and charge. Spin is an intrinsic property of a particle manifested in having permanent magnetic moment. A particle with spin can interact with magnetic fields just like particles with charge can interact with electric fields. This is why spin is pictorially depcited as tiny manget.

The Schrodinger equation does not account for electron spin. The concept of electron spin originates from Dirac’s relativistic equation. However, it can be included in the Schrodinger equation as an extra quantum number (\(s\)). Furthermore, it appears to follow the general laws of angular momentum. The spin angular momentum vector \(\vec{S}\) has a magnitude: \(|\vec{S}| = S = \sqrt{s(s+1)}\hbar\) where \(s\) is the spin quantum number (\(\frac{1}{2}\)).

Eigenvalues of spin#

To summarize the experiments desicovering electron spin we can write down eigenvalues of spin. We know only two values are observed in the experiments hence we must have two eigenfunctions and two eigenvalues.
We use analogy with the angular momentum to write downt the two expressions for the mangitude and projection of spin.

\[{S^2 = s(s+1)\hbar^2 = \frac{3}{4}\hbar^2\textnormal{ (since }s = \frac{1}{2}\textnormal{)}}\]

\[{S_z = m_s\hbar\textnormal{ with }m_s = \pm\frac{1}{2}}\]

Eigenfunctions of spin#

The corresponding operators are denoted by \(\hat{S}_z\) and \(\hat{S}^2\). How about the eigenfunctions? The eigenfunctions are denoted by \(\alpha\) and \(\beta\) and we don’t write down their specific forms. The following relations apply for these eigenfunctions:

\[{\hat{S}^2\alpha\equiv \hat{S}^2|\alpha\rangle = \frac{1}{2}\left(\frac{1}{2} + 1\right)\hbar^2\alpha = \frac{3}{4}\hbar^2\alpha\equiv\frac{3}{4}\hbar^2|\alpha\rangle}\]

\[{\hat{S}^2\beta\equiv \hat{S}^2|\beta\rangle = \frac{1}{2}\left(\frac{1}{2} + 1\right)\hbar^2\beta = \frac{3}{4}\hbar^2\beta\equiv\frac{3}{4}\hbar^2|\beta\rangle}\]

\[{\hat{S}_z\alpha\equiv \hat{S}_z|\alpha\rangle = +\frac{1}{2}\hbar\alpha\equiv +\frac{1}{2}\hbar |\alpha\rangle}\]

\[{\hat{S}_z\beta\equiv \hat{S}_z|\beta\rangle = -\frac{1}{2}\hbar\beta\equiv -\frac{1}{2}\hbar |\beta\rangle}\]

Note that all the following operators commute: \(\hat{H}\), \(\hat{L}^2\), \(\hat{L}_z\), \(\hat{S}^2\), and \(\hat{S}_z\). This means that they all can be specified simultaneously. The spin eigenfunctions are taken to be orthonormal:

\[{\int\alpha^*\alpha d\sigma\equiv\langle\alpha|\alpha\rangle = \int\beta^*\beta d\sigma\equiv\langle\beta|\beta\rangle = 1}\]

\[{\int\alpha^*\beta d\sigma\equiv\langle\alpha|\beta\rangle = \int\beta^*\alpha d\sigma\equiv\langle\beta|\alpha\rangle = 0}\]

where the integrations are over variables that the spin eigenfunctions depend on. Note that we have not specified the actual forms these eigenfunctions. We have only stated that they follow from the rules of angular momentum. A complete wavefunction for a hydrogen like atom must specify also the spin part. The total wavefunction is then a product of the spatial wavefunction and the spin part.

Mangetic moment due to spin#

Note that analogously, the \(\hat{S}_x\) and \(\hat{S}_y\) operators can be defined. These do not commute with \(\hat{S}_z\). Because electrons have spin angular momentum, the unpaired electrons in silver atoms (Stern-Gerlach experiment) produce an overall magnetic moment (``the two two spots of silver atoms’’). The spin magnetic moment is proportional to its spin angular momentum:

\[{\vec{\hat{\mu}}_S = -\frac{g_ee}{2m_e}\vec{\hat{S}}}\]

where \(g_e\) is the free electron \(g\)-factor (2.002322 from quantum electrodynamics). The \(z\)-component of the spin magnetic moment is (\(z\) is the quantiziation axis):

\[{\hat{\mu}_z = -\frac{g_ee}{2m_e}\hat{S}_z}\]

Hamiltonian depends on spin operator#

Following the eigenfunctions of \(S_z\) obtained above the corresponding eigenvalues for spin mangetic moment will be:

\[{\mu_z = -\frac{g_ee\hbar}{2m_e}m_s = -g_e\mu_Bm_s}\]

Thus the total energy for a spin in an external magnetic field is:

\[{E = g_e\mu_Bm_sB}\]

where \(B\) is the magnetic field strength (in Tesla). By combining the contributions from the hydrogenlike atom Hamiltonian and the orbital and electron Zeeman terms, we have the total Hamiltonian:

\[{\hat{H} = \hat{H}_0 + \frac{eB}{2m_e}\hat{L}_z + \frac{g_eeB}{2m_e}\hat{S}_z = \hat{H}_0 + \frac{eB}{2m_e}\left(\hat{L}_z + g_e\hat{S}_z\right)}\]

The eigenvalues of this operator are (derivation not shown):

\[{E_{n,m_l,m_s} = -\frac{m_ee^4Z^2}{2(2\pi\epsilon_0)^2\hbar n^2} + \frac{eB\hbar}{2m_e}\left(m_l + g_em_s\right)}\]

Spin-Orbit coupling#

Having two source of mangeitc fields in atoms one due to orbtial momentum and another due to spin there arises a possibility that these microscopic magnets can interact. And such a possibility is indeed realized and known under name of spin-orbit coupling! A new term is added to hamitlonian to account for this fact.

\[\hat{H} = \hat{H}_0 + \frac{A}{r^3} \cdot \hat{L}\hat{S}\]

Where the A is a constant, \(\hat{H}_0\) hamiltonain for H-atom without spin and \(\hat{L}\) and \(\hat{S}\) are operators for total angular and spin momentum. Notice two things here: one, the term decays as \(1/r^3\) that is faster than potential energy hence this term makes much smaller contribution compared to hamiltonian for H-atom \(\hat{H}_0\). Secondly becasue of spin-orbit coupling neither \(\hat{L}\) nor \(hat{S}\) commute with hamiltonian! Hence one needs a new quantum number to specify H-atom states. this new quantum number is total angular momentum \(J+L+S\)

spin-ornit coupling — Fig. 40 Determining the values of total angular momentum \(j=l+s\) by using the fact that projections \(m_j\) are given as scalar sum of projections of angular and spin quantum numbers \(m_j=m_l+m_s\) and that \(j\) values follow the anuglar momentum quantization and assume \(2j+1\) values. Shown is the example of determining microstates for \(l=1\) state of H-atom (\(2p^1\)) orbital which gives rise to \(^2P_{3/2}\) and \(^2P_{1/2}\) microstates.#

Becsue of spint orbit coupling the energy levels are no longer diescribed by \(l\) and \(s\) separetely. This is why one introduces term sybols to describe new states with total spin multiplicity \((2S+1)\) and anuglar momentum \(L\) and total angular momentum \(J=L+S\). The word total will take more meaning when we discuss multi electorn atoms where angular moenta are summed. For H-atom \(S=1/2\) and \(L=l\)

\[^{2S+1}L_J\]

Example:

Determine the microstates of l=1 and l=2 resulting from spin-orbit coupling.

Solution

The \(j=l+s\) values follow the rules of spatial quantization of angular momentum just like \(l\) and \(s\). That is the projection of \(J\), the m_j can only take \(l+s, l+s-1,...l-s\) values. Since \(s=1/2\) we are left with \(l+1/2\) and \(l-1/2\) values of \(m_j\). Note that the total number of microstates is equal to 2 values of \(m_s\) \(+1/2, -1/2\) and \(2l+1\) values of \(m_l\). In total we expect \(2(2l+1)\) microstates.
For l=1 we identify \(m_j=1+1/2=3/2\) the maximum projection which is coming from j=3/2 valyes. We also idenitfy \(m_j=l-1/2=1/2\) the minimum projection which should come from \(j=1/2\)
We thus have two microstates

“Anomalous” Zeeman Effect#

While the Zeeman effect in some atoms (e.g., hydrogen) showed the expected equally-spaced triplet, in other atoms the magnetic field split the lines into four, six, or even more lines and some triplets showed wider spacings than expected. These deviations were labeled the anomalous Zeeman effect and were very puzzling to early researchers.
The explanation of these different patterns of splitting gave additional insight into the effects of electron spin. With the inclusion of electron spin in the total angular momentum, the other types of multiplets formed part of a consistent picture. So what has been historically called the “anomalous” Zeeman effect is really the normal Zeeman effect when electron spin is included.

When magnetic field is applied to \(H\) atom the external field \(B\) now interacts with the total combined angular momentum \(J\) which results in splitting of energy values of each level with distinct \(J\). That is \(^2P_{3/2}\) is split into \(2\cdot(3/2+1)=5\) lines and \(^2P_{1/2}\) is split into \(2(1/2+1)=3\) lines.

anomalus zeeman — Fig. 42 Anomalous Zeeman effect. (left) The spectral lines are split into multiple microstates becasue of spin-orbit coupling. Energy states are described by term symbols (right) With weak magnetic field the lines are split further acording to values of \(j\) and its projection \(m_j\).#

Selection rules#

Just like with energetic transitions of other model systems for H-atom probability of transition is given by transition moment or its z-projection \(\langle n,l,m_l, m_s |z|n,l,m_l, m_s \rangle\). Evaluating the expression using properties of special functions we get the following selection rules

\[\Delta L=\pm 1\]

\[\Delta S= 0\]

\[\Delta J=0, \pm 1\,\,(0\rightarrow 0\,\, forbidden)\]

Summary of spin and angular momentum#

Spin emerges naturally once one accounts for relativistic effect, as was originally shown by Paul Dirac. Except for special cases relativisti effects however are not too significant to include in quantum mechanics therefore we incoprorate spin as an additional degree of freedom which has not been accounted for but which is knwon to exist!

Angular momentum	Spin momentum
\(\hat{L}=\hat{r}\times \hat{p}\) \(\hat{L}_z = -i\hbar \Big (x\frac{\partial}{\partial y}-y\frac{\partial}{\partial x} \Big)\)	\(\hat{S}\) \(\hat{S}_z\)
\(l=0,1,2,3,...\) \(m_l=-l...0...l\)	\(s=1/2\) \(m_s=-1/2,1/2\)
\(\mid l,m_l\rangle=Y_{l, m_l}\)	\(\mid s,m_s\rangle=\alpha,\beta\) \(\mid 1/2,+ 1/2\rangle=\alpha\) \(\mid 1/2,- 1/2\rangle=\beta\)
\(L=\hbar\sqrt{l(l+1)}\) \(L_z=\hbar m\)	\(S=\hbar\sqrt{s(s+1)}=\hbar\sqrt{3/4}\) \(S_z=\hbar m_s= \pm \hbar/2\)
\(\mu_L=- g_l \frac{e}{2m_e}L\) \(g_l=1\)	\(\mu_S = g_s \frac{e}{2m_e}S\) \(g_s \approx 2\)

Problems#

Problem-1#

Why is it sufficient to specify \(|s, m_s\rangle\) for spin states

Problem-2#

Why we use four quantum numbers \(n\), \(l\), \(m_l\), \(m_s\) to specify H-atom states \(|1, 0, 0, +1/2\rangle\) for instance.
What can we say about operators whose eigenvalues are defined in terms of these quantum numbers?

Problem-3#

Determine microstates of H-atom correpsonding to \(l=4\)