Quantum Mechanics

Table of Contents

1. Formalism

1.1. Hilbert Space

Set of all possible quantum states \(\ket{\psi}\) is postulated to form a Hilbert space \(\mathcal{H}\).

The state space \(\mathcal{H}\) of many-aspect system is the tensor product of the state spaces of each aspect \(\mathcal{H}_i\). \[ \mathcal{H} = \bigotimes_i\mathcal{H}_i. \]

1.2. Operator

It is postulated that for every observable \(Q\) there exists hermitian operator \(\hat{Q}\) such that: \[ \ev{Q} = \mel{\psi}{\hat{Q}}{\psi}. \] furthermore, for eigenstates \(\ket{Q_i}\), \[ Q_i\ket{Q_i} = \hat{Q} \ket{Q_i}. \]

1.2.1. Momentum Operator

Momentum operator \(\hat{\mathbf{p}}\) is the generator of translation \(\hat{U}(\mathbf{x})\): \[ \hat{U}(\mathbf{x}) = e^{-i\hat{\mathbf{p}}\vdot \mathbf{x}/\hbar}. \]

\begin{align*} \grad \hat{U}(\mathbf{x})\ket{\mathbf{x}} &= \left( \frac{\hat{U}(\dd{\mathbf{x}}) - \hat{U}(\mathbf{0})}{\dd{\mathbf{x}}} \right)\ket{\mathbf{x}} \\ &= \frac{\ket{\mathbf{x} + \dd{\mathbf{x}}} - \ket{\mathbf{x}}}{\dd{\mathbf{x}}} \\ &= \pdv{\mathbf{x}} \ket{\mathbf{x}} \\ &= \frac{1}{i\hbar} \hat{\mathbf{p}} \ket{\mathbf{x}}. \end{align*}

The position basis form of momentum operator is derived as follows:

\begin{align*} \mel{\mathbf{x}}{\hat{\mathbf{p}}}{\psi} &= \bra{\mathbf{x}} \int_{\Omega} \dd[3]{\mathbf{x}'} \psi(\mathbf{x}')\hat{\mathbf{p}}\ket{\mathbf{x}'} \\ &= \bra{\mathbf{x}} \int_{\Omega} \dd[3]{\mathbf{x}'} \psi(\mathbf{x}')i\hbar \pdv{\mathbf{x}'}\ket{\mathbf{x}'} \\ &= \bra{\mathbf{x}} \int_{\Omega} \dd[3]{\mathbf{x}'}i\hbar \pdv{\mathbf{x}'}\left( \psi(\mathbf{x}')\ket{\mathbf{x}'} \right) + \bra{\mathbf{x}} \int_{\Omega} \dd[3]{\mathbf{x}'} \left( - i\hbar \pdv{\mathbf{x}'}\psi(\mathbf{x}') \right)\ket{\mathbf{x}'}\\ &= \bra{\mathbf{x}}\oint_{\partial \Omega}\dd[2]{\mathbf{x}'}i\hbar \psi(\mathbf{x}')\ket{\mathbf{x}'} + \bra{\mathbf{x}} \int_{\Omega} \dd[3]{\mathbf{x}'} \left( - i\hbar \pdv{\mathbf{x}'}\psi(\mathbf{x}') \right)\ket{\mathbf{x}'} \\ &= \bra{\mathbf{x}} \int_{\Omega} \dd[3]{\mathbf{x}'} \left( - i\hbar \pdv{\mathbf{x}'}\psi(\mathbf{x}') \right)\ket{\mathbf{x}'} \\ &= \int_{\Omega} \dd[3]{\mathbf{x}'} \left( - i\hbar \pdv{\mathbf{x}'}\psi(\mathbf{x}') \right)\delta^3(\mathbf{x} - \mathbf{x}') \\ &= -i\hbar \pdv{\mathbf{x}} \psi(\mathbf{x}) \end{align*}

where the wave function \(\psi(\mathbf{x})\) is assumed to be localized and the boundary term vanishes.

The form of position basis momentum operator looks natural because we know from de Broglie \(\mathbf{p} = \hbar \mathbf{k}\), and differentiation operator would pull that out. Plane wave is considered here in the spirit of Fourier transform.

It is similar for the energy operator.

1.2.2. Spectral Theory

The eigenstates are complete and orthonormal basis of the entire state space for a specific problem. Orthonorality comes from the Hermiticity of the observable.

Each eigenstate is paired with eigenvalue

  • The eigenvalues are the only allowed values upon observation.
  • The eigenstates can be superposed with the coefficient representing the probability amplitude, but when measured it will collapse into one of the eigenstates.

Eigenfunction

  • The eigenstate can be projected to a basis, then the resulting eigenvalue equation can be solved instead.
  • But the projection into the position space does not capture the spin of the particle.

1.2.3. Lie Theory

The unitary operators \(\hat{U}\) form a Lie group, and the Hermitian operators \(\hat{H}\) are related to the Lie algebra that generate that Lie group by the exponential map: \[ \exp\colon \frac{1}{i\hbar}\hat{H} \mapsto \hat{U}. \]

Hermitian operator of an observable \(p\) maps to transformation in its conjugate variable \(q\). The justification comes from the quantum equivalent of Hamilton's equations:

\begin{align*} &\pdv{q}\ev{H} = - \dv{t} \ev{p} \\ \implies &\pdv{\bra{\psi}}{q} \hat{H}\ket{\psi} + \bra{\psi} \pdv{\hat{H}}{q}\ket{\psi} + \bra{\psi}\hat{H} \pdv{\ket{\psi}}{q}= - \left(\dv{\bra{\psi}}{t} \hat{p}\ket{\psi} + \bra{\psi} \dv{\hat{p}}{t}\ket{\psi} + \bra{\psi}\hat{p} \dv{\ket{\psi}}{t}\right). \end{align*}

Let us assume that the Hamiltonian operator does not explicitly depend on \(q\) (not \(\hat{q}\)), and canonical momentum operator \(\hat{p}\) does not explicitly depend on time. Using Schrödinger Equation, the equation becomes:

\begin{align*} \pdv{\bra{\psi}}{q} \hat{H}\ket{\psi} + &\bra{\psi}\hat{H} \pdv{\ket{\psi}}{q} \\ = \mel{\psi}{-\frac{1}{i\hbar}\hat{p}\hat{H}}{\psi} + &\mel{\psi}{\hat{H}\frac{1}{i\hbar}\hat{p}}{\psi}. \end{align*}

This strongly suggests the following identity \[ i\hbar \pdv{q} \ket{\psi} = \hat{p} \ket{\psi} \] and consequently the generator relation: \[ \hat{U}(q) = \exp(\frac{\hat{p} q}{i\hbar}). \] where \(\hat{U}(q)\) is the translation operator in the general coordinate \(q\) by \(q\) amount.

Similar argument can be made for the second Hamilton equation. From \[ \pdv{p}\ev{H} = \dv{t}\ev{q}, \] we get: \[ -i\hbar \pdv{p}\ket{\psi} = \hat{q}\ket{\psi} \] and \[ \hat{U}(p) = \exp(-\frac{\hat{q}p}{i\hbar}) \]

1.3. Canonical Commutation Relation

\[ \frac{1}{i\hbar}[\hat{\mathbf{x}}, \hat{\mathbf{p}}] = \mathbf{1}. \]

It is easy to show from the assumption that momentum is the generator of spatial translation: \[ \ev{\left[\hat{\mathbf{x}}, \frac{1}{i\hbar}\hat{\mathbf{p}}\right]} = \pdv{\ev{x}}{x} = 1. \]

1.4. Uncertainty Relation

  • Uncertainty Principle

The uncertainty in \(A\) is given by \[ \sigma_A := \sqrt{\ev{A^2} - \ev{A}^2}. \]

Let us define \(\ket{f} = \ket{\hat{A} - \ev{A}}\) and \(\ket{f} = \ket{\hat{A} - \ev{A}}\). By Cauchy-Schwarz inequiality \[ \braket{f}\braket{g} \ge |\left\langle \braket{f}{g} \right\rangle|^2 \] After evaluating the operator and ignoreing the real part of the right hand side, \[ \sigma_A\sigma_B \ge \frac{1}{2i}\ev{[\hat{A}, \mathbf{B}]}^2. \]

This answers why the electron doesn't fall to the nucleus. Quantum particles cannot be localized, because of quantum fuzziness. By uncertainty principle, if a particle happens to be localized, then it would have very high momentum for the particle to fly away.

1.5. Born's Rule

The probability of state \(|\alpha\rangle\) collapsing to the state \(|\beta\rangle\) is given by: \[ | \braket{\beta}{\alpha} | ^2 \]

\(\langle \uparrow | \downarrow \rangle = 0\), according to the Stern-Gerlach Experiment, therefore \(|\uparrow\rangle\) and \(|\downarrow\rangle\) are orthogonal. This requires the magnitude of the state to be 1.

Imaginary amplitude now have physical significance since the probability, which we measure, depends on it.

1.5.1. New Born's Rule

\[ \frac{|\langle \beta | \alpha \rangle|^2\hskip-4pt}{\langle \beta|\beta\rangle\langle\alpha|\alpha\rangle} \] This allows the magnitude of a state to be other than 1.

1.6. Physical Equivalence

\(\forall z \in \mathbb{C}, \mathbf{u}\equiv_{ph} z\mathbf{u}\). Since the length of spin state does not matter, we normalize the spin state so that it has length \(1\).

The difference of phase is what matters, not individual phases. This eventually leads to U(1) gauge symmetry.

2. Schrödinger Equation

Quantum equivalent of Hamilton's equations.

The basis-independent equation is given by: \[ i\hbar \pdv{t} \ket{\psi} = \hat{H}|\psi\rangle \] where \(\hat{H}\) is the hamiltonian operator, and \(\ket{\psi}\) is the state vector.

The solution to this equation can be given simply as: \[ \ket{\psi(t)} = e^{-i\hat{H}t/\hbar} \ket{\psi_0}. \]

2.1. Basis-dependent Equations

Position-basis Equation \[ i\hbar \pdv{t}\psi(\mathbf{x}) = \mel{\mathbf{x}}{\hat{H}}{\psi} \]

For a simple Hamiltonian of the form \(H = T + V\), \[ i\hbar \pdv{t}\psi(\mathbf{x}) = -\frac{\hbar^2}{2m}\laplacian\psi(\mathbf{x}) + V(\mathbf{x})\psi(\mathbf{x}) \]

Momentum-basis Equation \[ i\hbar \pdv{t}\phi(p) = \mel{p}{\hat{H}}{\psi} \]

For simple Hamiltonian of the form \(H = T + V\), \[ i\hbar \pdv{t} \phi(p) = \frac{p^2}{2m}\phi(p) + V \left( i\hbar \pdv{p} \right)\phi(p). \]

Energy-basis Equation \[ i\hbar \pdv{t}c(E) = \mel{E}{\hat{H}}{\psi} \]

For simple Hamiltonian of the form \(H = T + V\), \[ i\hbar \pdv{t} c(E) = E c(E). \]

2.2. Time-Independent Schrödinger Equation

When \(\psi\) is of the form \(\psi(\mathbf{x})e^{-iEt/\hbar}\) , we have \[ \hat{H}\psi = E\psi \] where \(\psi(\mathbf{x})e^{-iEt/\hbar}\) is called the energy eigenstate.

2.3. For Hydrogen Atom

The potential energy operator is given by the electrostatic potential energy: \[ \hat{V} = -\frac{e^2}{4\pi\varepsilon_0\hat{r}}. \]

The solution can be obtained using the separation of variable technique in spherical coordinates: \[ \psi=R_{n,l}(r)Y_l^m(\theta, \phi)e^{-iEt/\hbar} \] where \(R_{n,l}(r)\) is radial function, \(Y_l^m(\theta, \phi)\) is sperical harmonics.

2.3.1. Radial Function

  • Starts positive at \(r=0\) , flips sign \(n-l-1\) times.
  • flattens as \(n\) gets large.

2.3.2. Sperical Harmonics

  • basis function for all sperical functions.
  • smooth at every points.
  • it is the orbitals in chemistry when the real part of the function is taken.
  • it flips its phase \(l-|m|\) times, when going from north pole to the south.
  • it rotates in phase \(m\) times, as circling along the azimuthal plane.

3. Heisenberg Picture

The state vector does not change, but the operator itself changes over time.

3.1. Heisenberg's Equation

  • Heisenberg Equation of Motion

\[ \dv{t} A_{\rm H}(t) = \frac{1}{i\hbar} [A_{\rm H}(t), H_{\rm H}(t)] + \left( \dv{A_{\rm S}}{t} \right)_{\rm H} \] where \(A_{\rm S}\) is the operator in Schrödinger picture, and subscript \(\rm H\) makes it into an operator in Heisenberg picture again.

Partial derivative notation is usually prefered for \(A_{\rm S}\) to emphasize the derivative is taken with respect to the explicit dependence on time of the operator. However, from mathematical point of view there is no reason to use it. I find it more confusing to use partial derivative, and I chose to stick to the mathematical convention.

4. Density Matrix

It can contain both quantum and statistical uncertainties For pure state \(\ket{\psi}\):

  • \[ \rho = \ket{\psi}\bra{\psi} \]
  • idempotent: \(\rho = \rho^2\)
  • purity of one: \(\tr \rho^2 = 1\)

For mixed state:

  • \[ \rho = \sum_j p_j \ket{\psi_j}\bra{\psi_j}\]

The expectation value can be calculated: \[ \expval{A} = \tr(\rho A). \]

4.1. Von Neumann Equation

  • Liouville-von Neumann equation

\[ i\hbar \dv{t}\rho = [H, \rho]. \]

5. Spin

Spin state is described by a \(2s+1\) dimensional complex vector, and they transform as spinor.

5.1. Stern-Gerlach Experiment

The silver atoms deflect based on the electron's spin state.

5.2. Bloch Sphere

5.2.1. Definition

Bloch sphere is a set \(\{(\theta,\phi)\mid 0\le\theta\le\pi, 0\le\phi\le 2\pi\}\), such that each element \((\theta, \phi)\) corresponds to a physically equivalent class of spin state \[ \begin{pmatrix} \cos(\theta/2)e^{i\phi_1} \\ \sin(\theta/2)e^{i\phi_2} \end{pmatrix},\ \text{where}\ \phi_1-\phi_2=\phi \] \(\theta\) is the elevation angle starts from the north pole. \(\phi\) is the azimuthal angle. This way, it makes correspondences: \[ (\theta=0, \phi)\leftrightarrow\begin{pmatrix} 1\\0 \end{pmatrix},\ (\theta=\pi, \phi)\leftrightarrow\begin{pmatrix} 0\\1 \end{pmatrix} \]

The spin state can be represented as a superposition of two state—/spin up/ and spin down in the \(z\) direction:

\begin{align*} |\mathord{+}x\rangle &= \frac{1}{\sqrt{2}}|\mathord{+}z\rangle + \frac{1}{\sqrt{2}}|\mathord{-}z\rangle \\ |\mathord{-}x\rangle &= \frac{1}{\sqrt{2}}|\mathord{+}z\rangle - \frac{1}{\sqrt{2}}|\mathord{-}z\rangle \\ |\mathord{+}y\rangle &= \frac{1}{\sqrt{2}}|\mathord{+}z\rangle + \frac{i}{\sqrt{2}}|\mathord{-}z\rangle \\ |\mathord{-}y\rangle &= \frac{1}{\sqrt{2}}|\mathord{+}z\rangle - \frac{i}{\sqrt{2}}|\mathord{-}z\rangle . \end{align*}

which then can be organized on the Bloch sphere.

5.2.2. Properties

The Poincaré sphere represents the polarization space which is abstraction of the real wave form, on the other hand the Bloch sphere visualize the space of spinors.

Riemann sphere represents the equivalence classes of the spinors under the physical equivalence, which is the Bloch sphere.

5.3. Pauli Spin Matrices

\begin{align*} \sigma_1 = \sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \\ \sigma_2 = \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} \\ \sigma_3 = \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \\ \end{align*}

Sometimes the identity matrix is included as \(\sigma_0\).

Pauli matrices are:

  • Hermitian, involutory, and unitary: \(\sigma = \sigma^{\dagger} = \sigma^{-1}\)
  • \(\{\sigma_0, \sigma_1, \sigma_2, \sigma_3\}\) is the basis for the real vector space of \(2\times 2\) Hermitian matrices.
  • anti-commute with each other: \(\sigma_i\sigma_j = -\sigma_j\sigma_i\)

It is the linear map (order 3 tensor) from the three dimensional vector space \(V\) to the spinor-pair space \(S\otimes S^*\): \[ \sigma_i\vphantom{\sigma}^{a}\vphantom{\sigma}_b \] where \(a\) is the index for the spinor space \(S\) and \(b\) is the index for the dual spinor space \(S^*\) in standard spinor basis.

They are also the three basis vector of the Clifford algebra \(\mathrm{Cl}_{3,0}(\mathbb{R})\).

5.3.1. Infeld-Van der Waerden Symbols

\[ \sigma_\mu^{a\dot{b}} \]

Generalization of the Pauli matrices to the four dimensional spacetime. It is written with Van der Waerden notation.

5.4. Spinor

5.4.1. Pauli Spinor

The factorization of the Pauli vectors into column and row spinors.

This is a "square root of a vector"

  • A vector is linearly mapped to a Pauli vector which then can be factored.
  • They multiply to form a rank 1 tensor, hence it is a rank ½ tensor.

The direct factorization is possible only when the determinant is zero \[ \begin{bmatrix} z & x-yi \\ x+yi & -z \end{bmatrix} = \begin{bmatrix} \sqrt{x-yi} \\ -i\sqrt{x+yi} \end{bmatrix} \begin{bmatrix} i\sqrt{x+yi} & \sqrt{x-yi} \end{bmatrix} =: \begin{bmatrix} \xi^1 \\ \xi^2 \end{bmatrix} \begin{bmatrix} -\xi^2 & \xi^1 \end{bmatrix} \] When the rank is greater than 1, the matrix can be written as the sum of product of spinors.

5.4.1.1. Rotation

It is rotated by a single \(\mathrm{SU}(2)\) matrix, which is half of the rotation of the Pauli vector, that does not change the length of the spinor: \[ \lVert U\xi \rVert^2 = (U\xi)^\dagger (U\xi) = \xi^\dagger U^\dagger U\xi = \xi^\dagger \xi = \lVert \xi \rVert^2. \]

The spinors on the Poincaré sphere and the Bloch sphere is also rotated in the same way.

5.4.1.2. Pauli Vector
  • Vectors with the basis of

\[ \begin{bmatrix}x\\ y \\ z \end{bmatrix} \rightsquigarrow \begin{bmatrix} z & x - yi \\ x+ yi & -z \end{bmatrix} = x\sigma_x + y\sigma_y + z\sigma_z \]

It is mostly equivalent to the three dimensional geometric algebra, with the exception that it admits null vector, since the components can be complex numbers.

Pauli vectors are

  • Traceless Hermitian Matrix
  • It squares to the square magnitude of the vector.
  • The determinant is the negative of the square magnitude.
  • The sum of pairs of sigma matrices are equivalent to the quaternion.
  • Both the group of unit quaternions and \(\mathrm{SU}(2)\) are isomorphic to the \(\mathrm{Spin}(3)\) Group, describing the spin in three dimensional space, which is the double cover of rotation group \(\mathrm{SO}(3)\).

Conjugation rotates \(\pi\) radians around the axis:

\begin{align*} \sigma_x &\rightsquigarrow \sigma_z\sigma_x\sigma_z^{-1} = \sigma_z\sigma_x\sigma_z = -\sigma_z\sigma_z\sigma_x = -\sigma_x \\ \sigma_y &\rightsquigarrow \sigma_z\sigma_y\sigma_z^{-1} = \sigma_z\sigma_y\sigma_z = -\sigma_z\sigma_z\sigma_y = -\sigma_y \\ \sigma_z &\rightsquigarrow \sigma_z\sigma_z\sigma_z^{-1} = \sigma_z\sigma_z\sigma_z = +\sigma_z \end{align*}

and negative conjugation by \(\sigma_z\) reflects along the \(z\) axis.

They can be used to reflect and rotate Pauli vectors

  • Two reflections is a rotation.
  • Pauli vectors are reflected by the Pauli matrices by the conjugation.
  • And rotated by the two pairs of unit Pauli vectors, which are the members of the \(\mathrm{SU}(2)\) group.
  • Using some abstract algebra, the \(\mathrm{SU}(2)\) matrices are all the matrix the rotates a Pauli vector, (up to a complex multiple, which does not matter what that is so the determinant is just taken to be 1).

5.4.2. Weyl Spinor

  • Spinor in four dimensional spacetime.

Weyl vector can be factored into Weyl spinors \[ \begin{bmatrix} ct + z & x-yi \\ x+yi & ct-z \end{bmatrix} = \begin{bmatrix} \psi^1 \\ \psi^2\end{bmatrix}\begin{bmatrix} {\psi^1}^* & {\psi^2}^*\end{bmatrix} = \begin{bmatrix} \psi^1 \\ \psi^2\end{bmatrix}\begin{bmatrix} \psi^{\dot{1}}& \psi^{\dot{2}}\end{bmatrix} \] with \(|\psi^1| = \sqrt{ct+z}, |\psi^2| = \sqrt{ct-z}\) and the phase difference \(\theta_2 - \theta_1 = \arctan(y/x)\).

5.4.2.1. Inner Product
  • \(\psi^\dagger \epsilon \phi\) is never conserved.
  • So \(\psi^\top \epsilon \phi\) is chosen as the inner product with
    • \[ \epsilon = \begin{bmatrix} 0 & 1 \\ -1 & 0\end{bmatrix}. \]
  • It is alternating.
  • \(\epsilon\) is called a symplectic form.
5.4.2.2. Dual Spinor

From the inner product \[ \psi^\top\epsilon \] the complex conjugate of a left spinor \(\psi\) becomes the right dual spinor \(\psi^*\).

The right spinor would be \(\psi^\dagger \epsilon\).

5.4.2.3. Chirality

It has left and right chirality, which is notated by \(\psi^a, \psi_a\) and \(\psi^{\dot{a}}, \psi_{\dot{a}}\).

This happens because the \(\mathrm{SO}(2,\mathbb{C})\) has two different representation, left and right. The two are connected by complex conjugation. Unlike \(\mathrm{SU}(2)\), there is no invertible matrix \(P\), such that \(L^* = P^{-1}LP\).

5.4.2.3.1. Van der Waerden Notation

Placing dot above the right chiral spinor indices.

5.4.2.4. Weyl Vector

Using Pauli spin matrices \[ ct\sigma_t + x\sigma_x + y\sigma_y + z\sigma_z = \begin{bmatrix} ct + z & x-yi \\ x+yi & ct-z \end{bmatrix} \] with \(\sigma_t\) being the identity matrix.

They are

  • Hermitian matrix
  • The determinant is the spacetime interval.
  • It transforms with \(\mathrm{SL}(2, \mathbb{C})\) matrices.
    • The transformation matrix is special and Hermitian
    • The boosts uses the hyperbolic functions

5.4.3. Dirac Spinor

It is direct sum of a left chiral Weyl spinor and a right chiral Weyl spinor. The SL(2, C) representation is: \[ \left( \frac{1}{2}, 0 \right) \oplus \left( 0, \frac{1}{2} \right). \]

5.4.4. Spin Group

The double cover of \(\mathrm{SO}(n)\) is called the spin group \(\mathrm{Spin}(n)\), which is the generalization of unit quaternions for \(n\)-dimensional space. The double cover of \(\mathrm{SO}^+(p,q)\) is called the spin group \(\mathrm{Spin}(p,q)\).

In general, spin groups is the minimal left ideal in Clifford algebra \(\mathrm{Cl}(p,q)\).

5.5. Spin-Statistics Theorem

  • Spin-Statistics Connection

The relationship between the intrinsic spin and the quantum particle statistics. It proves the observed statistics.

5.5.1. Statement

Integral spin particles are bosons with Bose-Einstein statistics, and half-integral spin particles are fermions with Fermi-Dirac statistics.

5.6. Spin-Orbit Coupling

When spin angular momentum \(\mathbf{S}\) and orbital angular momentum \(\mathbf{L}\) are coupled, the total angular momentum \(\mathbf{J} = \mathbf{L} + \mathbf{S}\) becomes more useful.

The tensor product space of 3D spinor space of p-orbital and 2D spinor space of spin can be decomposed into 4D spinor space and 2D spinor space under representation of Lorentz transformation: \[ 1 \otimes \frac{1}{2} = \frac{3}{2} \oplus \frac{1}{2}. \] That leaves us with four degenerate microstates \(2p_{3/2}\) and two degenerate microstates \(2p_{1/2}\).

6. Electromagnetic Interaction

6.1. Electric Field

The hamiltonian of electron bound to an atom and subjected to a constant (or approximately constant) electric field \(\mathbf{E}\), the hamiltonian becomes \[ \hat{H} = \hat{H}_0 + Q\phi_{\rm ext}(\mathbf{r}_0) - \hat{Q}^i E_i(\mathbf{r}_0) -\frac{1}{2}\hat{Q}^{ij}\partial_iE_j(\mathbf{r}_0) + \cdots. \] Approximate it up to dipole, and use perturbation on the time-independent Schrödinger equation,

6.2. Aharonov-Bohm Effect

The isolated presence of the magnetic field still can affect the motion of a charged particle, since the magnetic vector potential might not be zero.

6.3. Pauli Equation

  • Schrödinger-Pauli Equation

Electromagnetic interaction for spin-1/2 particle. \[ i\hbar\pdv{t}\ket{\psi} = \left[ \frac{1}{2m} (\vb*{\sigma}\vdot (\hat{\mathbf{p}} - q\mathbf{A}))^2 + q\phi\right]\ket{\psi} \] where \(\vb*{\sigma} = (\sigma_1, \sigma_2, \sigma_3)\), \(q\) is electric charge, \(\mathbf{A}\) is the magnetic vector potential.

While for scalar particle \[ i\hbar \frac{\partial \psi}{\partial t} =\frac{(\hat{\mathbf{p}} - q\mathbf{A})^2}{2m}\psi + \hat{V}\psi. \]

7. Wigner Quasipropability Distribution

  • Wigner function, Wigner-Ville distribution
  • The "probability" over phase space

\[ W(x,p) := \frac{1}{\pi \hbar} \int_{-\infty}^{\infty} \psi^{*}(x+y) \psi(x-y) e^{2ipy/\hbar} \dd{y} \]

Short-term Fourier transform with itself as the weight function.

8. Relativity

8.1. Classical

\(\ket{\psi(t)}\) is a function of time, the time is given as a parameter, and the time operator does not exists in non-relativistic setting.

8.2. Relativistic Quantum Mechanics

\[ [X^\mu, P^\nu] = \eta^{\mu\nu}i\hbar \]

8.3. Quantum Field Theory

\[ \Psi(x,y,z,t) \]

See quantum fields theory.

9. References

Author: Jeemin Kim

Created: 2026-10-27 Tue 05:57