This article summarizes equations in the theory of quantum mechanics.
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.
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Quantity (common name/s) |
(Common) symbol/s |
Defining equation |
SI unit |
Dimension |
Wavefunction |
ψ, Ψ |
To solve from the Schrödinger equation |
varies with situation and number of particles |
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Wavefunction probability density |
ρ |
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m−3 |
[L]−3 |
Wavefunction probability current |
j |
Non-relativistic, no external field:
star * is complex conjugate |
m−2⋅s−1 |
[T]−1 [L]−2 |
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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.
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Property or effect |
Nomenclature |
Equation |
Wavefunction for N particles in 3d |
- r = (r1, r2... rN)
- sz = (sz 1, sz 2, ..., sz N)
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In function notation:
in bra–ket notation:
for non-interacting particles:
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Position-momentum Fourier transform (1 particle in 3d) |
- Φ = momentum–space wavefunction
- Ψ = position–space wavefunction
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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.
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General normalization condition |
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Wave–particle duality and time evolution
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Property or effect |
Nomenclature |
Equation |
Planck–Einstein equation and de Broglie wavelength relations |
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Schrödinger equation |
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General time-dependent case:
Time-independent case:
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Heisenberg equation |
- Â = operator of an observable property
- [ ] is the commutator
- denotes the average
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Time evolution in Heisenberg picture (Ehrenfest theorem) |
of a particle. |
For momentum and position;
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Non-relativistic time-independent Schrödinger equation
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.
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One particle |
N particles |
One dimension |
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where the position of particle n is xn. |
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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]
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for non-interacting particles
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Three dimensions |
where the position of the particle is r = (x, y, z). |
where the position of particle n is r n = (xn, yn, zn), and the Laplacian for particle n using the corresponding position coordinates is
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for non-interacting particles
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Non-relativistic time-dependent Schrödinger equation
Again, summarized below are the various forms the Hamiltonian takes, with the corresponding Schrödinger equations and forms of solutions.
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One particle |
N particles |
One dimension |
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where the position of particle n is xn. |
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Three dimensions |
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This last equation is in a very high dimension,[2] so the solutions are not easy to visualize. |
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Photoemission
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Property/Effect |
Nomenclature |
Equation |
Photoelectric equation |
- Kmax = Maximum kinetic energy of ejected electron (J)
- h = Planck constant
- f = frequency of incident photons (Hz = s−1)
- φ, Φ = Work function of the material the photons are incident on (J)
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Threshold frequency and Work function |
- φ, Φ = Work function of the material the photons are incident on (J)
- f0, ν0 = Threshold frequency (Hz = s−1)
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Can only be found by experiment.
The De Broglie relations give the relation between them:
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Photon momentum |
- p = momentum of photon (kg m s−1)
- f = frequency of photon (Hz = s−1)
- λ = wavelength of photon (m)
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The De Broglie relations give:
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Quantum uncertainty
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Property or effect |
Nomenclature |
Equation |
Heisenberg's uncertainty principles |
- n = number of photons
- φ = wave phase
- [, ] = commutator
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Position–momentum
Energy-time
Number-phase
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Dispersion of observable |
A = observables (eigenvalues of operator) |
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General uncertainty relation |
A, B = observables (eigenvalues of operator) |
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Angular momentum
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Property or effect |
Nomenclature |
Equation |
Angular momentum quantum numbers |
- s = spin quantum number
- ms = spin magnetic quantum number
- ℓ = Azimuthal quantum number
- mℓ = azimuthal magnetic quantum number
- j = total angular momentum quantum number
- mj = total angular momentum magnetic quantum number
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Spin:
Orbital:
Total:
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Angular momentum magnitudes |
angular momementa:
- S = Spin,
- L = orbital,
- J = total
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Spin magnitude:
Orbital magnitude:
Total magnitude:
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Angular momentum components |
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Spin:
Orbital:
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- Magnetic moments
In what follows, B is an applied external magnetic field and the quantum numbers above are used.
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