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* {{cite book|title=Electromagnetism (2nd Edition)|author=I.S. Grant, W.R. Phillips, Manchester Physics|publisher=John Wiley & Sons|year=2008|isbn=978-0-471-92712-9}} |
* {{cite book|title=Electromagnetism (2nd Edition)|author=I.S. Grant, W.R. Phillips, Manchester Physics|publisher=John Wiley & Sons|year=2008|isbn=978-0-471-92712-9}} |
||
* {{cite book|title=Introduction to Electrodynamics|edition=3rd |author=D.J. Griffiths|publisher=Pearson Education, Dorling Kindersley, |year=2007|isbn=81-7758-293-3}} |
* {{cite book|title=Introduction to Electrodynamics|edition=3rd |author=D.J. Griffiths|publisher=Pearson Education, Dorling Kindersley, |year=2007|isbn=81-7758-293-3}} |
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* {{cite book| author=Arundhati chandrashekhar Gatole, Arundhati chandrashekhar Gatole, M.J. Hodgeson| title=Essential Principles of Physics| publisher=John Murray|edition=2nd| year=1978 | isbn=0-7195-3382-1}} |
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==Further reading== |
==Further reading== |
Revision as of 17:05, 23 September 2013
Part of a series of articles about |
Quantum mechanics |
---|
This article summarizes equations in the theory of quantum mechanics.
Definitions
Quantity (common name/s) | (Common) symbol/s | Defining equation | SI units | Dimension |
---|---|---|---|---|
Wave function | ψ, Ψ | To solve from the Schrödinger equation | varies with situation and number of particles | |
Threshold frequency | f0, ν0 | Can only be found by experiment/calculated from work function below. | Hz = s−1 | [T]−1 |
Work function | φ, Φ | J | [M][L]2[T]−2 | |
Wavefunction probability density | ρ | 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 |
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 descrete 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 necerssary). Following are general mathematical results, used in calculations.
Property or effect | Nomenclature | Equation |
---|---|---|
Wavefunction for N particles in 3d |
|
In function notation:
in bra-ket notation: for non-interacting particles:
|
Position-momentum Fourier transform (1 particle in 3d) |
|
|
General probability distribution |
|
|
General normalization condition |
Equations
Property/Effect | Nomenclature | Equation |
---|---|---|
Photoelectric equation | ||
photon momentum | ||
Cutoff wavelength |
Wave–particle duality and time evolution
Property or effect | Nomenclature | Equation |
---|---|---|
Planck–Einstein equation and de Broglie wavelength relations |
|
|
Schrödinger equation |
|
General time-dependent case:
Time-independent case: |
Heisenberg equation |
|
|
Time evolution in Heisenberg picture (Ehrenfest theorem) |
of a particle. |
For momentum and position;
|
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.
One particle | N particles | |
One dimension |
where the position of particle n is xn. | |
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] |
for non-interacting particles
| |
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
|
for non-interacting particles
|
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.
One particle | N particles | |
One dimension |
where the position of particle n is xn. | |
Three dimensions | ||
This last equation is in a very high dimension,[2] so the solutions are not easy to visualize. | ||
Quantum uncertainty
Property or effect | Nomenclature | Equation |
---|---|---|
Heisenberg's uncertainty principles |
|
Position-momentum
Energy-time Number-phase |
Dispersion of observable |
|
|
General uncertainty relation |
|
Property or effect | Nomenclature | Equation |
---|---|---|
Density of states | ||
Fermi-Dirac distribution (fermions) |
|
|
Bose-Einstein distribution (bosons) |
Angular momentum
Property or effect | Nomenclature | Equation |
---|---|---|
Angular momentum quantum numbers |
|
Spin projection:
Orbital:
Total: |
Angular momentum magnitudes | angular momementa:
|
Spin magnitude:
Orbital magnitude: Total magnitude:
|
Angular momentum components | Spin:
Orbital: |
- 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 |
|
z-component: |
spin magnetic dipole moment |
|
z-component: |
dipole moment potential |
|
The Hydrogen atom
Property or effect | Nomenclature | Equation |
---|---|---|
Energy levels |
|
|
Spectrum | λ = wavelength of emitted photon, during electronic transition from Ei to Ej |
See also
- Defining equation (physical chemistry)
- List of equations in classical mechanics
- List of equations in wave theory
- List of relativistic equations
- List of equations in fluid mechanics
- List of equations in gravitation
- List of electromagnetism equations
- List of photonics equations
- List of equations in nuclear and particle physics
Footnotes
- ^
Feynman, R.P.; Leighton, R.B.; Sand, M. (24). "Operators". The Feynman Lectures on Physics. Vol. 3. Addison-Wesley. pp. 20–7. ISBN 0-201-02115-3.
{{cite book}}
: Check date values in:|date=
and|year=
/|date=
mismatch (help); Unknown parameter|month=
ignored (help); Unknown parameter|yesr=
ignored (help) - ^ Shankar, R. (1994). Principles of Quantum Mechanics. Kluwer Academic/Plenum Publishers. p. 141. ISBN 978-0-306-44790-7.
Sources
- 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 9-781429-202657.
- L.N. Hand, J.D. Finch (2008). Analytical Mechanics. Cambridge University Press,. ISBN 978-0-521-57572-0.
{{cite book}}
: CS1 maint: extra punctuation (link) - T.B. Arkill, C.J. Millar (1974). Mechanics, Vibrations and Waves. John Murray,. ISBN 0-7195-2882-8.
{{cite book}}
: CS1 maint: extra punctuation (link) - H.J. Pain (1983). The Physics of Vibrations and Waves (3rd ed.). John Wiley & Sons,. ISBN 0-471-90182-2.
{{cite book}}
: CS1 maint: extra punctuation (link) - J.R. Forshaw, A.G. Smith (2009). Dynamics and Relativity. Wiley,. ISBN 978-0-470-01460-8.
{{cite book}}
: CS1 maint: extra punctuation (link) - 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.
{{cite book}}
: CS1 maint: multiple names: authors list (link) - D.J. Griffiths (2007). Introduction to Electrodynamics (3rd ed.). Pearson Education, Dorling Kindersley,. ISBN 81-7758-293-3.
{{cite book}}
: CS1 maint: extra punctuation (link) - Arundhati chandrashekhar Gatole, Arundhati chandrashekhar Gatole, M.J. Hodgeson (1978). Essential Principles of Physics (2nd ed.). John Murray. ISBN 0-7195-3382-1.
{{cite book}}
: CS1 maint: multiple names: authors list (link)
Further reading
- 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.