List of equations in quantum mechanics

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This article summarizes equations in the theory of quantum mechanics.

Wavefunctions[edit]

A fundamental physical constant occurring in quantum mechanics is the Planck constant, h. A common abbreviation is ħ = h/2π, also known as the reduced Planck constant or Dirac constant.

Quantity (common name/s) (Common) symbol/ Defining equation SI units Dimension
Wave function ψ, Ψ To solve from the Schrödinger equation varies with situation and number of particles
Wavefunction probability density ρ \rho = \left | \Psi \right |^2 = \Psi^* \Psi m−3 [L]−3
Wavefunction probability current j Non-relativistic, no external field:

\mathbf{j} = \frac{-i\hbar}{2m}\left(\Psi^* \nabla \Psi - \Psi \nabla \Psi^*\right)  = \frac\hbar m \mathrm{Im}(\Psi^*\nabla\Psi)=\mathrm{Re}(\Psi^* \frac{\hbar}{im} \nabla \Psi)

star * is complex conjugate

m−2 s−1 [T]−1 [L]−2

The general form of wavefunction for a system of particles, each with position ri and z-component of spin sz i. Sums are over the discrete variable sz, integrals over continuous positions r.

For clarity and brevity, the coordinates are collected into tuples, the indices label the particles (which cannot be done physically, but is mathematically necessary). Following are general mathematical results, used in calculations.

Property or effect Nomenclature Equation
Wavefunction for N particles in 3d
  • r = (r1, r2... rN)
  • sz = (sz 1, sz 2... sz N)
In function notation:

 \Psi  = \Psi \left (\mathbf{r}, \mathbf{s_z}, t \right )

in bra–ket notation:  |\Psi\rangle = \sum_{s_{z1}} \sum_{s_{z2}}\cdots\sum_{s_{zN}}\int_{V_1}\int_{V_2}\cdots\int_{V_N} \mathrm{d}\mathbf{r}_1\mathrm{d}\mathbf{r}_2\cdots\mathrm{d}\mathbf{r}_N \Psi |\mathbf{r}, \mathbf{s_z}\rangle

for non-interacting particles:

 \Psi  = \prod_{n=1}^N\Psi \left (\mathbf{r}_n,s_{zn}, t \right )

Position-momentum Fourier transform (1 particle in 3d)
  • Φ = momentum-space wavefunction
  • Ψ = position-space wavefunction
\begin{align} \Phi(\mathbf{p},s_z,t) & = \frac{1}{\sqrt{2\pi\hbar}} \int\limits_{\mathrm{all \, space}} e^{-i\mathbf{p}\cdot\mathbf{r}/\hbar} \Psi(\mathbf{r}, s_z,t)\mathrm{d}^3\mathbf{r} \\ 
&\upharpoonleft \downharpoonright\\
\Psi(\mathbf{r},s_z,t) & = \frac{1}{\sqrt{2\pi\hbar}} \int\limits_{\mathrm{all \, space}} e^{+i\mathbf{p}\cdot\mathbf{r}/\hbar} \Phi(\mathbf{r},s_z,t)\mathrm{d}^3\mathbf{p}_n \\
\end{align}
General probability distribution
  • Vj = volume (3d region) particle may occupy,
  • P = Probability that particle 1 has position r1 in volume V1 with spin sz1 and particle 2 has position r2 in volume V2 with spin sz2, etc.
 P = \sum_{s_{zN}}\cdots\sum_{s_{z2}}\sum_{s_{z1}}\int_{V_N}\cdots\int_{V_2}\int_{V_1} \left | \Psi \right |^2\mathrm{d}^3\mathbf{r}_1\mathrm{d}^3\mathbf{r}_2\cdots\mathrm{d}^3\mathbf{r}_N\,\!
General normalization condition  P = \sum_{s_{zN}}\cdots\sum_{s_{z2}}\sum_{s_{z1}}\int\limits_{\mathrm{all \, space}}\cdots\int\limits_{\mathrm{all \, space}}\int\limits_{\mathrm{all \, space}} \left | \Psi \right |^2\mathrm{d}^3\mathbf{r}_1\mathrm{d}^3\mathbf{r}_2\cdots\mathrm{d}^3\mathbf{r}_N = 1\,\!

Equations[edit]

Wave–particle duality and time evolution[edit]

Property or effect Nomenclature Equation
Planck–Einstein equation and de Broglie wavelength relations
\mathbf{P} = (E/c, \mathbf{p}) = \hbar(\omega /c ,\mathbf{k}) = \hbar \mathbf{K}
Schrödinger equation
General time-dependent case:

i\hbar\frac{\partial}{\partial t} \Psi = \hat{H}\Psi

Time-independent case: \hat{H}\Psi = E\Psi

Heisenberg equation
  • Â = operator of an observable property
  • [ ] is the commutator
  • \langle \, \rangle denotes the average
\frac{d}{dt}\hat{A}(t)=\frac{i}{\hbar}[\hat{H},\hat{A}(t)]+\frac{\partial \hat{A}(t)}{\partial t},
Time evolution in Heisenberg picture (Ehrenfest theorem)

of a particle.

\frac{d}{dt}\langle \hat{A}\rangle = \frac{1}{i\hbar}\langle [\hat{A},\hat{H}] \rangle+ \left\langle \frac{\partial \hat{A}}{\partial t}\right\rangle

For momentum and position;

m\frac{d}{dt}\langle \mathbf{r}\rangle = \langle \mathbf{p} \rangle

\frac{d}{dt}\langle \mathbf{p}\rangle = -\langle \nabla V \rangle

Non-relativistic time-independent Schrödinger equation[edit]

Summarized below are the various forms the Hamiltonian takes, with the corresponding Schrödinger equations and forms of wavefunction solutions. Notice in the case of one spatial dimension, for one particle, the partial derivative reduces to an ordinary derivative.

One particle N particles
One dimension  \hat{H} = \frac{\hat{p}^2}{2m} + V(x) = -\frac{\hbar^2}{2m}\frac{d^2}{d x^2} + V(x)  \begin{align}\hat{H} &= \sum_{n=1}^{N}\frac{\hat{p}_n^2}{2m_n} + V(x_1,x_2,\cdots x_N) \\ 
& = -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\frac{\partial^2}{\partial x_n^2} + V(x_1,x_2,\cdots x_N) 
\end{align}

where the position of particle n is xn.

 E\Psi = -\frac{\hbar^2}{2m}\frac{d^2}{d x^2}\Psi + V\Psi  E\Psi = -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\frac{\partial^2}{\partial x_n^2}\Psi + V\Psi \, .
 \Psi(x,t)=\psi(x) e^{-iEt/\hbar} \, .

There is a further restriction — the solution must not grow at infinity, so that it has either a finite L2-norm (if it is a bound state) or a slowly diverging norm (if it is part of a continuum):[1] \| \psi \|^2 = \int |\psi(x)|^2\, dx.\,

 \Psi = e^{-iEt/\hbar}\psi(x_1,x_2\cdots x_N)

for non-interacting particles

 \Psi = e^{-i{E t/\hbar}}\prod_{n=1}^N\psi(x_n) \, , \quad V(x_1,x_2,\cdots x_N) = \sum_{n=1}^N V(x_n) \, .

Three dimensions  \begin{align}\hat{H} & = \frac{\hat{\mathbf{p}}\cdot\hat{\mathbf{p}}}{2m} + V(\mathbf{r}) \\
& = -\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r}) 
\end{align}

where the position of the particle is r = (x, y, z).

 \begin{align} \hat{H} & = \sum_{n=1}^{N}\frac{\hat{\mathbf{p}}_n\cdot\hat{\mathbf{p}}_n}{2m_n} + V(\mathbf{r}_1,\mathbf{r}_2,\cdots\mathbf{r}_N) \\
& = -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\nabla_n^2 + V(\mathbf{r}_1,\mathbf{r}_2,\cdots\mathbf{r}_N) 
\end{align}

where the position of particle n is r n = (xn, yn, zn), and the Laplacian for particle n using the corresponding position coordinates is

\nabla_n^2=\frac{\partial^2}{{\partial x_n}^2} + \frac{\partial^2}{{\partial y_n}^2} + \frac{\partial^2}{{\partial z_n}^2}

 E\Psi = -\frac{\hbar^2}{2m}\nabla^2\Psi + V\Psi  E\Psi = -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\nabla_n^2\Psi + V\Psi
 \Psi = \psi(\mathbf{r}) e^{-iEt/\hbar}  \Psi = e^{-iEt/\hbar}\psi(\mathbf{r}_1,\mathbf{r}_2\cdots \mathbf{r}_N)

for non-interacting particles

 \Psi = e^{-i{E t/\hbar}}\prod_{n=1}^N\psi(\mathbf{r}_n) \, , \quad V(\mathbf{r}_1,\mathbf{r}_2,\cdots \mathbf{r}_N) = \sum_{n=1}^N V(\mathbf{r}_n)

Non-relativistic time-dependent Schrödinger equation[edit]

Again, summarized below are the various forms the Hamiltonian takes, with the corresponding Schrödinger equations and forms of solutions.

One particle N particles
One dimension  \hat{H} = \frac{\hat{p}^2}{2m} + V(x,t) = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x,t)  \hat{H} = \sum_{n=1}^{N}\frac{\hat{p}_n^2}{2m_n} + V(x_1,x_2,\cdots x_N,t)  
= -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\frac{\partial^2}{\partial x_n^2} + V(x_1,x_2,\cdots x_N,t)

where the position of particle n is xn.

 i\hbar\frac{\partial}{\partial t}\Psi = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}\Psi + V\Psi  i\hbar\frac{\partial}{\partial t}\Psi = -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\frac{\partial^2}{\partial x_n^2}\Psi + V\Psi \, .
 \Psi = \Psi(x,t)  \Psi = \Psi(x_1,x_2\cdots x_N,t)
Three dimensions  \begin{align}\hat{H} & = \frac{\hat{\mathbf{p}}\cdot\hat{\mathbf{p}}}{2m} + V(\mathbf{r},t) \\
& = -\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r},t) \\
\end{align}  \begin{align} \hat{H} & = \sum_{n=1}^{N}\frac{\hat{\mathbf{p}}_n\cdot\hat{\mathbf{p}}_n}{2m_n} + V(\mathbf{r}_1,\mathbf{r}_2,\cdots\mathbf{r}_N,t) \\
& = -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\nabla_n^2 + V(\mathbf{r}_1,\mathbf{r}_2,\cdots\mathbf{r}_N,t) 
\end{align}
 i\hbar\frac{\partial}{\partial t}\Psi = -\frac{\hbar^2}{2m}\nabla^2\Psi + V\Psi  i\hbar\frac{\partial}{\partial t}\Psi = -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\nabla_n^2\Psi + V\Psi

This last equation is in a very high dimension,[2] so the solutions are not easy to visualize.

 \Psi = \Psi(\mathbf{r},t)  \Psi = \Psi(\mathbf{r}_1,\mathbf{r}_2,\cdots\mathbf{r}_N,t)

Photoemmission[edit]

Property/Effect Nomenclature Equation
Photoelectric equation
  • Kmax = Maximum kinetic energy of ejected electron (J)
  • h = Planck's constant
  • f = frequency of incident photons (Hz = s−1)
  • φ, Φ = Work function of the material the photons are incident on (J)
K_\mathrm{max} = hf - \Phi\,\!
Threshold frequency and
  • φ, Φ = Work function of the material the photons are incident on (J)
  • f0, ν0 = Threshold frequency (Hz = s−1)
Can only be found by experiment.

The De Broglie relations give the relation between them:

\phi = hf_0\,\!

Photon momentum
  • p = momentum of photon (kg m s−1)
  • f = frequency of photon (Hz = s−1)
  • λ = wavelength of photon (m)

The De Broglie relations give:

p = hf/c = h/\lambda\,\!

Quantum uncertainty[edit]

Property or effect Nomenclature Equation
Heisenberg's uncertainty principles
  • n = number of photons
  • φ = wave phase
  • [, ] = commutator
Position-momentum

\sigma(x) \sigma(p) \ge \frac{\hbar}{2} \,\!

Energy-time \sigma(E) \sigma(t) \ge \frac{\hbar}{2} \,\!

Number-phase \sigma(n) \sigma(\phi) \ge \frac{\hbar}{2} \,\!

Dispersion of observable
  • A = observables (eigenvalues of operator)

\begin{align}
\sigma(A)^2 & = \langle(A-\langle A \rangle)^2\rangle \\
& = \langle A^2 \rangle - \langle A \rangle^2
\end{align}

General uncertainty relation
  • A, B = observables (eigenvalues of operator)
\sigma(A)\sigma(B) \geq \frac{1}{2}\langle i[\hat{A}, \hat{B}] \rangle
Probability Distributions
Property or effect Nomenclature Equation
Density of states N(E) = 8\sqrt{2}\pi m^{3/2}E^{1/2}/h^3\,\!
Fermi–Dirac distribution (fermions)
  • P(Ei) = probability of energy Ei
  • g(Ei) = degeneracy of energy Ei (no of states with same energy)
  • μ = chemical potential
P(E_i) = g(E_i)/(e^{(E-\mu)/kT}+1)\,\!
Bose–Einstein distribution (bosons) P(E_i) = g(E_i)/(e^{(E_i-\mu)/kT}-1)\,\!

Angular momentum[edit]

Property or effect Nomenclature Equation
Angular momentum quantum numbers
  • s = spin quantum number
  • ms = spin magnetic quantum number
  • = Azimuthal quantum number
  • m = azimuthal magnetic quantum number
  • mj = total angular momentum magnetic quantum number
  • j = total angular momentum quantum number
Spin projection:

m_s \in \{-s,-s+1\cdots s-1,s\}\,\!

Orbital: m_\ell \in \{-\ell,-\ell+1\cdots \ell-1,\ell\}\,\!
m_\ell \in \{0 \cdots n-1\}\,\!

Total: \begin{align}& j = \ell +s \\
& j \in \{|\ell-s|,|\ell-s|+1 \cdots |\ell+s|-1,|\ell+s| \} \\
\end{align}\,\!

Angular momentum magnitudes angular momementa:
  • S = Spin,
  • L = orbital,
  • J = total
Spin magnitude:

|\mathbf{S}| = \hbar\sqrt{s(s+1)}\,\!

Orbital magnitude: |\mathbf{L}| = \hbar\sqrt{\ell(\ell+1)}\,\!

Total magnitude: \mathbf{J} = \mathbf{L} + \mathbf{S}\,\!

|\mathbf{J}| = \hbar\sqrt{j(j+1)}\,\!

Angular momentum components Spin:

S_z = m_s \hbar\,\!

Orbital: L_z = m_\ell \hbar\,\!

Magnetic moments

In what follows, B is an applied external magnetic field and the quantum numbers above are used.

Property or effect Nomenclature Equation
orbital magnetic dipole moment
\boldsymbol{\mu}_\ell = -e\mathbf{L}/2m_e = g_\ell \frac{\mu_B}{\hbar} \mathbf{L}\,\!

z-component: \mu_{\ell,z} = -m_\ell\mu_B\,\!

spin magnetic dipole moment
\boldsymbol{\mu}_s = -e\mathbf{S}/m_e = g_s \frac{\mu_B}{\hbar} \mathbf{S}\,\!

z-component: \mu_{s,z} = -e S_z/m_e = g_seS_z/2m_e\,\!

dipole moment potential
  • U = potential energy of dipole in field
U = -\boldsymbol{\mu}\cdot\mathbf{B} = -\mu_z B\,\!

The Hydrogen atom[edit]

Main article: Hydrogen atom
Property or effect Nomenclature Equation
Energy levels
E_n = -me^4/8\epsilon_0^2h^2n^2 = 13.61eV/n^2\,\!
Spectrum λ = wavelength of emitted photon, during electronic transition from Ei to Ej \frac{1}{\lambda} = R\left(\frac{1}{n_j^2} - \frac{1}{n_i^2}\right), \, n_j<n_i\,\!

See also[edit]

Footnotes[edit]

  1. ^ Feynman, R.P.; Leighton, R.B.; Sand, M. (1964). "Operators". The Feynman Lectures on Physics 3. Addison-Wesley. pp. 20–7. ISBN 0-201-02115-3. 
  2. ^ Shankar, R. (1994). Principles of Quantum Mechanics. Kluwer Academic/Plenum Publishers. p. 141. ISBN 978-0-306-44790-7. 

Sources[edit]

  • P.M. Whelan, M.J. Hodgeson (1978). Essential Principles of Physics (2nd ed.). John Murray. ISBN 0-7195-3382-1. 
  • G. Woan (2010). The Cambridge Handbook of Physics Formulas. Cambridge University Press. ISBN 978-0-521-57507-2. 
  • A. Halpern (1988). 3000 Solved Problems in Physics, Schaum Series. Mc Graw Hill. ISBN 978-0-07-025734-4. 
  • R. G. Lerner, G. L. Trigg (2005). Encyclopaedia of Physics (2nd ed.). VHC Publishers, Hans Warlimont, Springer. pp. 12–13. ISBN 978-0-07-025734-4. 
  • C. B. Parker (1994). McGraw Hill Encyclopaedia of Physics (2nd ed.). McGraw Hill. ISBN 0-07-051400-3. 
  • P. A. Tipler, G. Mosca (2008). Physics for Scientists and Engineers: With Modern Physics (6th ed.). W. H. Freeman and Co. ISBN 978-1-4292-0265-7. 
  • L.N. Hand, J. D. Finch (2008). Analytical Mechanics. Cambridge University Press,. ISBN 978-0-521-57572-0. 
  • T. B. Arkill, C. J. Millar (1974). Mechanics, Vibrations and Waves. John Murray,. ISBN 0-7195-2882-8. 
  • H.J. Pain (1983). The Physics of Vibrations and Waves (3rd ed.). John Wiley & Sons,. ISBN 0-471-90182-2. 
  • J. R. Forshaw, A. G. Smith (2009). Dynamics and Relativity. Wiley,. ISBN 978-0-470-01460-8. 
  • G. A. G. Bennet (1974). Electricity and Modern Physics (2nd ed.). Edward Arnold (UK). ISBN 0-7131-2459-8. 
  • I. S. Grant, W. R. Phillips, Manchester Physics (2008). Electromagnetism (2nd Edition). John Wiley & Sons. ISBN 978-0-471-92712-9. 
  • D.J. Griffiths (2007). Introduction to Electrodynamics (3rd ed.). Pearson Education, Dorling Kindersley,. ISBN 81-7758-293-3. 
  • Arundhati chandrashekhar Gatole (2003–2004). Electrodynamics (5th ed.). Pearson Education, Dorling Kindersley. ISBN 978-0-521-57572 Check |isbn= value (help). 

Further reading[edit]

  • L. H. Greenberg (1978). Physics with Modern Applications. Holt-Saunders International W. B. Saunders and Co. ISBN 0-7216-4247-0. 
  • J. B. Marion, W. F. Hornyak (1984). Principles of Physics. Holt-Saunders International Saunders College. ISBN 4-8337-0195-2. 
  • A. Beiser (1987). Concepts of Modern Physics (4th ed.). McGraw-Hill (International). ISBN 0-07-100144-1. 
  • H. D. Young, R. A. Freedman (2008). University Physics – With Modern Physics (12th ed.). Addison-Wesley (Pearson International). ISBN 0-321-50130-6.