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Equation sheet

Units

Constants

Energy ConstantValue
h=6.63⋅10−34 [J⋅s]h = 6.63 \cdot 10^{-34} \, [J \cdot s]mu=1.66⋅10−27 [kg]m_u = 1.66 \cdot 10^{-27} \, [kg]
kB=1.38⋅10−23 [J⋅K−1]k_B = 1.38 \cdot 10^{-23} \, [J \cdot K^{-1}]NA=6.02⋅1023 [mol−1]N_A = 6.02 \cdot 10^{23} \, [mol^{-1}]
c=3.0⋅108 [m⋅s−1]c = 3.0 \cdot 10^{8} \, [m \cdot s^{-1}]me=9.1093837015⋅10−31 [kg]m_e = 9.1093837015 \cdot 10^{-31} \, [kg]
RH=109680 [cm−1]R_H = 109680 \, [cm^{-1}]e=1.602176634⋅10−19 [C]e = 1.602176634 \cdot 10^{-19} \, [C]

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From classical to Quantum

Blackbody radiation

DescriptionEquations
Energy quantizationE=nhνE = nh\nu
Average energy of an oscillating dipole⟨Eosc⟩=hνehν/kBT−1\langle E_{\text{osc}} \rangle = \dfrac{h\nu}{e^{h\nu / k_B T} - 1}
Spectral radiation density of blackbody (Planck)ρ(ν,T) dν=8πhν3c31ehν/kBT−1 dν\rho (\nu, T) \ d\nu = \dfrac{8\pi h \nu^3}{c^3}\dfrac{1}{e^{h\nu / k_B T} - 1} \ d\nu
Spectral radiation density of blackbody (classical, Rayleigh-Jeans)ρ(ν,T) dν=8πν2c3kBT dν\rho (\nu, T) \ d\nu = \dfrac{8\pi \nu^2}{c^3} k_B T \ d\nu

Wave-particle duality

DescriptionEquations
Energy of lightE=hνE = h \nu
Photoelectric effect
Kinetic energy of ejected photoelectron		Ek=hν−ΦE_k = h \nu - \Phi
de Broglie relationp=hλp = \dfrac{h}{\lambda}
Kinetic energyEk=12mv2=p22mE_k = \dfrac{1}{2}mv^2 = \dfrac{p^2}{2m}

Atomic spectra of hydrogen and Bohr’s model

DescriptionEquations
Hydrogen emission lines
n2>n1n_2 > n_1		ν~=1λ=RH(1n12−1n22)\tilde{\nu} = \dfrac{1}{\lambda} = R_H \left( \dfrac{1}{n_1^2} - \dfrac{1}{n_2^2} \right)
Bohr’s radiusr=4πε0ℏ2mee2r = \dfrac{4\pi \varepsilon_0 \hbar^2}{m_e e^2}
Energy level in Bohr’s modelEn=−mee48ε02h2n2E_n = -\dfrac{m_e e^4}{8\varepsilon_0^2 h^2n^2}
Emission of hydrogen atom
n2>n1n_2 > n_1		ν=mee48ε02h3(1n12−1n22)\nu = \dfrac{m_e e^4}{8\varepsilon_0^2 h^3} \left( \dfrac{1}{n_1^2} - \dfrac{1}{n_2^2} \right)

Waves

DescriptionEquations
Classical nondispersive wave equation∂2Ψ(x,t)∂x2=1v2∂2Ψ(x,t)∂t2\dfrac{\partial^2 \Psi(x, t)}{\partial x^2} = \dfrac{1}{v^2} \dfrac{\partial^2 \Psi(x, t)}{\partial t^2}
Wave numberk=2πλk = \dfrac{2\pi}{\lambda}
Frequencyν=1T\nu = \dfrac{1}{T}
Angular frequencyω=2πT=2πν\omega = \dfrac{2\pi}{T} = 2\pi\nu
Wave speedv=λνv = \lambda\nu
Euler’s formulaeiθ=cos⁡θ+isin⁡θe^{i\theta} = \cos\theta + i\sin\theta
Solution of wave equationΨ(x,t)=Asin⁡(kx−ωt+ϕ) =Re(Aei(kx−ωt+ϕ′))\begin{aligned}\Psi(x, t) &= A \sin(kx - \omega t + \phi) \\\ &= \mathrm{Re}(Ae^{i(kx-\omega t + \phi')})\end{aligned}
Interfering traveling waves give standing waveΨ(x,t)=A[sin⁡(kx−ωt)+sin⁡(kx+ωt)] =2Asin⁡(kx)cos⁡(ωt) =ψ(x)cos⁡(ωt)\begin{aligned}\Psi(x, t) &= A[\sin(kx - \omega t) + \sin(kx + \omega t)] \\\ &= 2A \sin(kx)\cos(\omega t) \\\ &= \psi(x)\cos(\omega t) \end{aligned}
Time-independent Schrodinger equation−ℏ22md2ψ(x)dx2+V(x)ψ(x)=Eψ(x)-\dfrac{\hbar^2}{2m}\dfrac{d^2\psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x)
Time-dependent Schrodinger equation−ℏ22m∂2Ψ(x,t)∂x2+V(x,t)Ψ(x,t)=iℏ∂Ψ(x,t)∂t-\dfrac{\hbar^2}{2m}\dfrac{\partial^2\Psi(x, t)}{\partial x^2} + V(x, t)\Psi(x, t) = i\hbar\dfrac{\partial \Psi(x, t)}{\partial t}
Stationary states are standing wavesΨ(x,t)=ψ(x)e−i(E/ℏ)t\Psi(x, t) = \psi(x) e^{-i(E/\hbar)t}
Normalization∥f(x)∥=∫Df∗f dx=1\Vert f(x) \Vert = \int_D f^* f \ dx = 1
Orthogonality∫Df∗g dx=0\int_D f^* g \ dx = 0
Use quantum mechanics when ...1. λparticle∼Lproblem\lambda_{\text{particle}} \sim L_{\text{problem}}
2. ΔE≳kbT\Delta E \gtrsim k_bT (discrete energy spectrum)

Quantum-Mechanical Postulates

  1. The state of a quantum-mechanical particle is completely specified by a wave function Ψ(x,t)\Psi(x, t). The probability that the particle will be found at time t0t_0 in a spatial interval of width dxdx centered at x0x_0 is given by Ψ∗(x0,t0)Ψ(x0,t0)dx\Psi^*(x_0, t_0)\Psi(x_0, t_0) dx

  2. For every measurable property of a system, there exists a corresponding operator.

  3. In any single measurement of the observable that corresponds to the operator A^\hat{A}, the only values that will ever be measured are the eigenvalues of that operator.

  4. If the system is in a state described by the wave function Ψ(x,t)\Psi(x, t), and the value of the observatle aa is measured once on each of many identically prepared systems, the average value (expectation value) of all of the measurements is given by

    ⟨a⟩=∫−∞∞Ψ∗A^Ψ dx∫−∞∞Ψ∗Ψ dx\langle a \rangle = \dfrac{\displaystyle\int_{-\infty}^{\infty} \Psi^* \hat{A} \Psi \ dx}{\displaystyle\int_{-\infty}^{\infty} \Psi^*\Psi \ dx}
    (1)

  5. The evolution in time of a quantum-mechanical system is governed by the time-dependent Schrödinger equation

    H^Ψ(x,t)=iℏ∂Ψ(x,t)∂t\hat{H}\Psi(x, t) = i\hbar\dfrac{\partial\Psi(x, t)}{\partial t}
    (2)

Operators

Description1D3D
Positionx^=x\hat{x} = xx^=x\mathbf{\hat{x}} = \mathbf{x}
Linear momentump^x=−iℏddx\hat{p}_x = -i\hbar \dfrac{d}{dx}p^=−iℏ∇\mathbf{\hat{p}} = -i\hbar\mathbf{\nabla}
Kinetic energyT^=p^x22m=−ℏ22md2dx2\hat{T} = \dfrac{\hat{p}_x^2}{2m} = -\dfrac{\hbar^2}{2m}\dfrac{d^2}{dx^2}T^=−ℏ22m∇2\mathbf{\hat{T}} = -\dfrac{\hbar^2}{2m}\mathbf{\nabla}^2
Potential energyV^=V(x)\hat{V} = V(x)V^=V(x)\mathbf{\hat{V}} = V(\mathbf{x})
Total energy HamiltonianH^=T^+V^=−ℏ22md2dx2+V(x)\hat{H} = \hat{T} + \hat{V} = -\dfrac{\hbar^2}{2m}\dfrac{d^2}{dx^2} + V(x)H^=−ℏ22m∇2+V(x)\mathbf{\hat{H}} = -\dfrac{\hbar^2}{2m}\mathbf{\nabla}^2 + V(\mathbf{x})

Simple Quantum Systems

Stationary states

DescriptionEquations
Time dependent Schrodinger equationH^Ψ(x,t)=iℏ∂Ψ(x,t)∂t\hat{H}\Psi(x, t) = i\hbar\dfrac{\partial\Psi(x, t)}{\partial t}
Time independent Schrodinger equationH^ψn(x)=Enψn(x)\hat{H}\psi_n(x) = E_n \psi_n(x)
Stationary state wave functionΨ(x,t)=ψ(x)T(t)\Psi(x, t) = \psi(x) T(t)
Time component of wave functionT(t)=e−iEt/ℏT(t) = e^{-iEt/\hbar}
Probability of finding particle in an intervalProb(x,x+dx)=∣Ψ(x,t)∣2dx=∣ψ(x)∣2dx\mathrm{Prob}(x, x+dx) = \vert \Psi(x, t) \vert^2 dx = \vert \psi(x) \vert^2 dx
General solution as linear combination of stationary statesψ(x)=∑ncnϕn(x)\psi(x) = \sum\limits_n c_n \phi_n(x)
Expansion coefficientscn=⟨ϕn∣ψ⟩=∫ϕn∗ψ dxc_n = \langle \phi_n \vert \psi \rangle = \int \phi_n^* \psi \ dx
Normalization$\sum\limits_n

Particle in a 1D box

DescriptionEquations
Time independent Schrodinger equation[−ℏ22md2dx2+V(x)]ψ(x)=Eψ(x)\left[ -\dfrac{\hbar^2}{2m}\dfrac{d^2}{dx^2} + V(x) \right] \psi(x) = E\psi(x)
Wave function
n=1,2,3,...n = 1, 2, 3, ...		ψn(x)=2Lsin⁡(nπxL)\psi_n(x) = \sqrt{\dfrac{2}{L}} \sin\left(\dfrac{n \pi x}{L}\right)
Energy eigenvaluesEn=h28mL2n2=ℏ2π22mL2n2E_n = \dfrac{h^2}{8mL^2} n^2 = \dfrac{\hbar^2 \pi^2}{2mL^2}n^2

Particle in a 3D box

DescriptionEquations
Time independent Schrodinger equation[−ℏ22m∇2+V(x)]ψ(x)=Eψ(x)\left[ -\dfrac{\hbar^2}{2m}\mathbf{\nabla}^2 + V(\mathbf{x}) \right] \psi(\mathbf{x}) = E\psi(\mathbf{x})
Wave function
nx=1,2,3,...n_x = 1, 2, 3, ...
ny=1,2,3,...n_y = 1, 2, 3, ...
nz=1,2,3,...n_z = 1, 2, 3, ...		ψnx,ny,nz(x) =ψnx(x)ψny(y)ψnz(z) =2Lx2Ly2Lzsin⁡(nxπxLx)sin⁡(nyπyLy)sin⁡(nzπzLz)\begin{aligned}&\psi_{n_x, n_y, n_z}(\mathbf{x}) \\\ =& \psi_{n_x}(x)\psi_{n_y}(y)\psi_{n_z}(z) \\\ =& \sqrt{\dfrac{2}{L_x}}\sqrt{\dfrac{2}{L_y}}\sqrt{\dfrac{2}{L_z}} \sin\left(\dfrac{n_x \pi x}{L_x}\right)\sin\left(\dfrac{n_y \pi y}{L_y}\right)\sin\left(\dfrac{n_z \pi z}{L_z}\right)\end{aligned}
Energy eigenvaluesEn=h28m(nx2Lx2+ny2Ly2+nz2Lz2)E_n = \dfrac{h^2}{8m} \left(\dfrac{n_x^2}{L_x^2} + \dfrac{n_y^2}{L_y^2} + \dfrac{n_z^2}{L_z^2}\right)

Finite potential well

DescriptionEquations
PotentialV(x)={0x∈[0,L] V0elsewhereV(x) = \begin{cases}0 & x\in [0, L] \\\ V_0 & \mathrm{elsewhere}\end{cases}
Reflection probabilityR=(E−E−V0)2(E+E−V0)2R = \dfrac{(\sqrt{E} - \sqrt{E - V_0})^2}{(\sqrt{E} + \sqrt{E - V_0})^2}
Transmission probabilityT=4E(E−V0)(E+E−V0)2T = \dfrac{4\sqrt{E(E - V_0)}}{(\sqrt{E} + \sqrt{E - V_0})^2}

Commutators and Uncertainty

DescriptionEquations
Commutator[A,B]=AB−BA[A, B] = AB - BA
Condition of commutation[A,B]=0[A, B] = 0
Standard deviation (uncertainty)σA=⟨(A−⟨A⟩)2⟩ =⟨A2⟩−⟨A⟩2\begin{aligned}\sigma_A &= \sqrt{\langle (A - \langle A \rangle)^2 \rangle} \\\ &= \sqrt{\langle A^2 \rangle - \langle A \rangle^2}\end{aligned}
Heisenberg uncertainty principle (general)σAσB≥12∣⟨[A^,B^]⟩∣\sigma_A \sigma_B \ge \frac{1}{2} \vert\langle[\hat{A}, \hat{B}]\rangle\vert
Heisenberg uncertainty principle (position-momentum)σxσp≥ℏ2\sigma_x \sigma_p \ge \frac{\hbar}{2}

Spectroscopy

Vibration: quantum harmonic oscillator

DescriptionEquations
Vibrational Schrodinger equation[−ℏ22μ1r2∂∂r(r2∂∂r)+V(r)]ψ(x)=Eψ(x)\left[ -\dfrac{\hbar^2}{2\mu}\dfrac{1}{r^2}\dfrac{\partial}{\partial r}\left(r^2 \dfrac{\partial}{\partial r}\right) + V(r) \right]\psi(\mathbf{x}) = E\psi(\mathbf{x})
Wave functionψ(x)=R(r)Y(θ,ϕ)\psi(\mathbf{x}) = R(r)Y(\theta, \phi)
Harmonic approximationV(r)≈12kr2V(r) \approx \frac{1}{2}kr^2
Spring constantk=μω02k = \mu\omega_0^2
Vibrational Schrodinger equation[−ℏ22μ1r2∂∂r(r2∂∂r)+12kr2]ψ(r)=Eψ(r)\left[ -\dfrac{\hbar^2}{2\mu}\dfrac{1}{r^2}\dfrac{\partial}{\partial r}\left(r^2 \dfrac{\partial}{\partial r}\right) + \dfrac{1}{2}kr^2 \right]\psi(r) = E\psi(r)
Wave function
n=0,1,2,...n = 0, 1, 2, ...		ψ(r)=12nn!(απ)1/4Hn(αr)e−αr2/2\psi(r) = \dfrac{1}{\sqrt{2^n n!}}\left(\dfrac{\alpha}{\pi}\right)^{1/4} H_n(\sqrt{\alpha} r) e^{-\alpha r^2 / 2}
Hermite polynomialsHn(x)=(−1)nex2(dndxn)e−x2H_n(x) = (-1)^n e^{x^2}\left(\dfrac{d^n}{dx^n}\right)e^{-x^2}
Constantα=mω0ℏ\alpha = \dfrac{m\omega_0}{\hbar}
Energy eigenvalue
n=0,1,2,...n = 0, 1, 2, ...		En=(n+12)ℏω0E_n = (n + \frac{1}{2})\hbar \omega_0
Transition dipole momentμ⃗fi=dμ⃗(x0)dx⟨ψf∣x^∣ψi⟩\vec{\mu}_{fi} = \dfrac{d\vec{\mu}(x_0)}{dx} \langle \psi_f \vert \hat{x} \vert \psi_i \rangle
Vibrational selection ruleΔn=±1\Delta n = \pm 1

Rotation: quantum rigid rotor

Classical rigid rotor

DescriptionEquations
Angular momentumL=x×p=Iω⃗\mathbf{L} = \mathbf{x} \times \mathbf{p} = I\vec{\omega}
Linear velocityv⃗=R0ω⃗\vec{v} = R_0 \vec{\omega}
Moment of inertiaI=mR02I = mR_0^2
Rotational kinetic energyE=12Iω2=L22IE = \dfrac{1}{2}I\omega^2 = \dfrac{L^2}{2I}

Quantum rigid rotor

DescriptionEquations
Angular momentum operatorL^=x^×p^\hat{\mathbf{L}} = \hat{\mathbf{x}} \times \hat{\mathbf{p}}
z-component of angular momentum operatorL^z=ℏi∂∂ϕ\hat{L}_z = \dfrac{\hbar}{i}\dfrac{\partial}{\partial\phi}
Magnitude of angular momentum operatorL^2=L2=−ℏ2[1sin⁡θ∂∂θ(sin⁡θ∂∂θ)+1sin⁡2θ∂2∂ϕ2]\hat{\mathbf{L}}^2 = L^2 = -\hbar^2 \left[ \dfrac{1}{\sin\theta}\dfrac{\partial}{\partial\theta} \left( \sin\theta\dfrac{\partial}{\partial\theta} \right) + \dfrac{1}{\sin^2\theta}\dfrac{\partial^2}{\partial\phi^2} \right]
Components of L^\hat{\mathbf{L}} does not commute[L^i,L^j]=iℏL^k[\hat{L}_i, \hat{L}_j] = i\hbar \hat{L}_k
Components of L^\hat{\mathbf{L}} commute with its magnitude[L^i,L2]=0[\hat{L}_i, L^2] = 0
DescriptionEquations
Rotational Schrodinger equation−ℏ22μR02[1r2sin⁡θ∂∂θ(sin⁡θ∂∂θ)+1r2sin⁡2θ∂2∂ϕ2]Y(θ,ϕ)=EY(θ,ϕ)-\dfrac{\hbar^2}{2\mu R_0^2}\left[ \dfrac{1}{r^2 \sin\theta}\dfrac{\partial}{\partial \theta}\left(\sin\theta\dfrac{\partial}{\partial \theta}\right) + \dfrac{1}{r^2 \sin^2 \theta}\dfrac{\partial^2}{\partial\phi^2} \right]Y(\theta, \phi) = EY(\theta, \phi)
Spherical harmonicsYlm(θ,ϕ)=(−)m(2l+1)4π(l−m)!(l+m)!Plm(cos⁡θ)eimϕY_l^m(\theta, \phi) = (-)^m \sqrt{\dfrac{(2l+1)}{4\pi}\dfrac{(l-m)!}{(l+m)!}} P_l^m(\cos\theta) e^{im\phi}
Legendre polynomialPlm(x)=12ll!(1−x2)m/2d(l+m)dx(l+m)(x2−1)lP_l^m(x) = \dfrac{1}{2^l l!}(1-x^2)^{m/2} \dfrac{d^{(l+m)}}{dx^{(l+m)}}(x^2-1)^l
Energy eigenvalues
l=0,1,2,...l = 0, 1, 2, ...		El=ℏ22Il(l+1)E_l = \dfrac{\hbar^2}{2I}l(l+1)
Angular momentum eigenvalues
l=0,1,2,...l = 0, 1, 2, ...		L2Y=ℏ2l(l+1)YL^2 Y = \hbar^2l(l+1) Y
z-component eigenvalues
m=−l,...,0,...,lm = -l, ..., 0, ..., l		LzY=ℏmYL_z Y = \hbar m Y
Transition dipole momentμfi=⟨ψf∣μzcos⁡θ∣ψi⟩\mu_{fi} = \langle \psi_f \vert \mu_z \cos\theta \vert \psi_i \rangle
Rotational selection ruleΔl=±1,Δm=0\Delta l = \pm 1, \Delta m = 0

Hydrogen atom

DescriptionEquations
Hydrogen atom Schrodinger equation[−ℏ22me1r2∂∂r(r2∂∂r)+L⃗22mer2−e2r]ψ(x)=Eψ(x)\left[ -\dfrac{\hbar^2}{2m_e}\dfrac{1}{r^2}\dfrac{\partial}{\partial r}\left(r^2 \dfrac{\partial}{\partial r}\right) + \dfrac{\vec{L}^2}{2m_er^2} - \dfrac{e^2}{r} \right]\psi(x) = E\psi(x)
Effective potentialVeff=ℏ2l(l+1)2mr2−e2rV_{\text{eff}} = \dfrac{\hbar^2 l(l+1)}{2mr^2} - \dfrac{e^2}{r}
Wave function
n=1,2,...n = 1, 2, ...		ψnlm(x)=Rnl(r)Ylm(θ,ϕ)\psi_{nlm}(x) = R_{nl}(r)Y_l^m(\theta, \phi)
Energy eigenvalues
n=1,2,...n = 1, 2, ...		En=−e22a01n2=−me42ℏ21n2=−RHn2E_n = -\dfrac{e^2}{2a_0}\dfrac{1}{n^2} = -\dfrac{me^4}{2\hbar^2}\dfrac{1}{n^2} = -\dfrac{R_H}{n^2}
Rydberg’s constantRH=2.179×10−18J=13.6 eVR_H = 2.179 \times 10^{-18} \mathrm{J} = 13.6 \ \mathrm{eV}
Bohr’s radiusa0=ℏ2me2a_0 = \dfrac{\hbar^2}{me^2}
Radial probability distributionPnl(r)dr=r2Rnl2(r)drP_{nl}(r) dr = r^2 R^2_{nl}(r) dr

Many Electron and Proton System

Many electron atom

DescriptionEquations
Helium Schrodinger equation[−ℏ22m∇12⏟_KE of e1−−ℏ22m∇22⏞KE of e2−ˆ−2e2∣x1∣⏟_e1−-N attraction−2e2∣x2∣⏞e2−-N attractionˆ+e2∣x1−x2∣⏟_e1−-e2−repulsion]ψ(x1,x2)=Eψ(x1,x2)\left[ \underbrace{-\dfrac{\hbar^2}{2m}\nabla_1^2}\_{\text{KE of }e^-_1} \overbrace{-\dfrac{\hbar^2}{2m}\nabla_2^2}\^{\text{KE of }e^-_2} \underbrace{-\dfrac{2e^2}{\vert\mathbf{x}_1\vert}}\_{e^-_1\text{-N attraction}} \overbrace{-\dfrac{2e^2}{\vert\mathbf{x}_2\vert}}\^{e^-_2\text{-N attraction}} + \underbrace{\dfrac{e^2}{\vert\mathbf{x}_1 - \mathbf{x}_2\vert}}\_{e^-_1 \text{-} e^-_2 \text{repulsion}} \right] \psi(\mathbf{x}_1, \mathbf{x}_2) = E \psi(\mathbf{x}_1, \mathbf{x}_2)
Orbital approximationψ(x1,x2,…,xn)=ϕ(x1)ϕ(x2)…ϕ(xn)\psi(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_n) = \phi(\mathbf{x}_1)\phi(\mathbf{x}_2)\dots\phi(\mathbf{x}_n)
Hartree orbital equations[−ℏ2∇i22m−Ze2∣x∣+∑j=1,j≠iN∫e2ϕj∗(x′)ϕj(x′)∣x−x′∣d3x′]ϕi(x)=εiiϕi(x)\left[ -\dfrac{\hbar^2 \nabla_i^2}{2m} - \dfrac{Ze^2}{\vert\mathbf{x}\vert} + \sum\limits^N_{j=1, j\not= i} \displaystyle\int \dfrac{e^2 \phi_j^*(\mathbf{x}')\phi_j(\mathbf{x}')}{\vert\mathbf{x} - \mathbf{x}' \vert} d^3\mathbf{x}' \right] \phi_i(\mathbf{x}) = \varepsilon_{ii} \phi_i(\mathbf{x})

Spin

DescriptionEquations
Components of S^\hat{\mathbf{S}} does not commute[S^i,S^j]=iℏS^k[\hat{S}_i, \hat{S}_j] = i\hbar \hat{S}_k
Components of S^\hat{\mathbf{S}} commute with its magnitude[S^i,S2]=0[\hat{S}_i, S^2] = 0
Eigenvalue of S^2\hat{\mathbf{S}}^2S^2↔ℏ2s(s+1)\hat{\mathbf{S}}^2 \leftrightarrow \hbar^2 s(s+1)
Eigenvalue of S^z\hat{S}_zS^z↔ℏms\hat{S}_z \leftrightarrow \hbar m_s

Electron spin

DescriptionEquations
Electron spins=12s = \frac{1}{2}
Spin up functionα(ms)={1ms=+12 0ms=−12\alpha(m_s) = \begin{cases} 1 & m_s = +\frac{1}{2} \\\ 0 & m_s = -\frac{1}{2} \end{cases}
Spin down functionβ(ms)={0ms=+12 1ms=−12\beta(m_s) = \begin{cases} 0 & m_s = +\frac{1}{2} \\\ 1 & m_s = -\frac{1}{2} \end{cases}
α\alpha is eigenfunction of Sz^\hat{S_z}Sz^α=+12ℏα\hat{S_z} \alpha = +\frac{1}{2}\hbar \alpha
β\beta is eigenfunction of Sz^\hat{S_z}Sz^β=−12ℏβ\hat{S_z} \beta = -\frac{1}{2}\hbar \beta
α,β\alpha, \beta are eigenfunctions of S2^\hat{S^2}S2^α=ℏ2s(s+1)α=34ℏ2αS2^β=ℏ2s(s+1)β=34ℏ2β\hat{S^2} \alpha = \hbar^2 s(s+1) \alpha = \frac{3}{4} \hbar^2 \alpha \newline \hat{S^2} \beta = \hbar^2 s(s+1) \beta = \frac{3}{4} \hbar^2 \beta
Normalization∑msα∗α=∑msβ∗β=1\sum\limits_{m_s} \alpha^*\alpha = \sum\limits_{m_s} \beta^*\beta = 1
Orthogonality∑msα∗β=∑msβ∗α=0\sum\limits_{m_s} \alpha^*\beta = \sum\limits_{m_s} \beta^*\alpha = 0

Identical particles

DescriptionEquations
Spin-spin permutation operatorPijψ(r1,r2,…,ri,…,rj,…,rN)=ψ(r1,r2,…,rj,…,ri,…,rN)P_{ij} \psi(\mathbf{r}_1, \mathbf{r}_2, \dots, \mathbf{r}_i, \dots, \mathbf{r}_j, \dots, \mathbf{r}_N) = \psi(\mathbf{r}_1, \mathbf{r}_2, \dots, \mathbf{r}_j, \dots, \mathbf{r}_i, \dots, \mathbf{r}_N)
Doing nothingPijPij=1P_{ij}P_{ij} = 1
Symmetric eigenvalueλ=1\lambda = 1
Anti-symmetric eigenvalueλ=−1\lambda = -1
Fermions (e.g. electron)12\frac{1}{2}-integer spin, anti-symmetric
Bosonsinteger spin, symmetric
Pauli exclusion principleψ(r1,r2,…,ri,…,ri,…,rN)=−ψ(r1,r2,…,ri,…,ri,…,rN)=0\psi(\mathbf{r}_1, \mathbf{r}_2, \dots, \mathbf{r}_i, \dots, \mathbf{r}_i, \dots, \mathbf{r}_N) = - \psi(\mathbf{r}_1, \mathbf{r}_2, \dots, \mathbf{r}_i, \dots, \mathbf{r}_i, \dots, \mathbf{r}_N) = 0
Slater determinantΨ(x1,x2,⋯ ,xN)=1N!∣χ1(x1)χ2(x1)⋯χN(x1) χ1(x2)χ2(x2)⋯χN(x2) ⋮⋮⋱⋮ χ1(xN)χ2(xN)⋯χN(xN)∣\Psi(\mathrm{x}_1, \mathrm{x}_2, \cdots, \mathrm{x}_N) = \dfrac{1}{\sqrt{N!}} \begin{vmatrix}\chi_1(\mathbf{x}_1) & \chi_2(\mathbf{x}_1) & \cdots & \chi_N(\mathbf{x}_1) \\\ \chi_1(\mathbf{x}_2) & \chi_2(\mathbf{x}_2) & \cdots & \chi_N(\mathbf{x}_2) \\\ \vdots & \vdots & \ddots & \vdots \\\ \chi_1(\mathbf{x}_N) & \chi_2(\mathbf{x}_N) & \cdots & \chi_N(\mathbf{x}_N) \end{vmatrix}
Hartree-Fock orbital equations[−ℏ2∇22m−Ze2∣x∣]ϕi(r)+∑j=1N[ϕi(r)∫e2ϕj∗(r)ϕj(r)∣x−x′∣d3r′−ϕj(r)∫e2ϕj∗(r)ϕi(r)∣x−x′∣d3r′]=εiϕi(r)\left[ -\dfrac{\hbar^2\nabla^2}{2m} - \dfrac{Ze^2}{\vert\mathbf{x}\vert} \right] \phi_i(\mathbf{r}) + \displaystyle\sum_{j=1}^N \left[ \phi_i(\mathbf{r}) \int \dfrac{e^2 \phi^*_j(\mathbf{r}) \phi_j(\mathbf{r})}{\vert\mathbf{x} - \mathbf{x}'\vert} d^3r' - \phi_j(\mathbf{r}) \int \dfrac{e^2 \phi^*_j(\mathbf{r}) \phi_i(\mathbf{r})}{\vert\mathbf{x} - \mathbf{x}'\vert} d^3r' \right] = \varepsilon_i \phi_i(\mathbf{r})
Molecular orbital by linear combination of atomic orbitals (MO-LCAO)ψ(x)=c1ϕ1(x)+c2ϕ2(x)MO=c1(AO)+c2(AO)\psi(\mathbf{x}) = c_1 \phi_1 (\mathbf{x}) + c_2 \phi_2 (\mathbf{x}) \newline \mathrm{MO} = c_1 (\mathrm{AO}) + c_2 (\mathrm{AO})
Variational principleE=⟨ψ∣H∣ψ⟩⟨ψ∣ψ⟩E = \dfrac{\langle \psi \vert H \vert \psi \rangle}{\langle \psi \vert \psi \rangle}
Appendix C: Python and Interactive Demos
Python Calculator