Lead Article: Tables of Physics Formulae
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Wave Particle Duality
edit
edit
Planck–Einstein Equation
E
=
h
f
=
h
c
λ
{\displaystyle E=hf={\frac {hc}{\lambda }}\,\!}
Photon Momentum
p
=
h
f
c
=
h
λ
{\displaystyle p={\frac {hf}{c}}={\frac {h}{\lambda }}\,\!}
De Broglie Wavelength
p
=
h
λ
=
ℏ
k
{\displaystyle p={\frac {h}{\lambda }}=\hbar k\,\!}
Heisenberg's Uncertainty
Principle
Δ
x
Δ
p
x
⩾
ℏ
2
{\displaystyle \Delta x\Delta p_{x}\geqslant {\frac {\hbar }{2}}\,\!}
Δ
E
Δ
t
⩾
ℏ
2
{\displaystyle \Delta E\Delta t\geqslant {\frac {\hbar }{2}}\,\!}
Photoelectric Effect :
Photons greater than threshold frequency incident on a metal surface
causes (photo)electrons to be emmited from surface.
E
k
m
a
x
=
h
f
−
Φ
{\displaystyle E_{\mathrm {k\;\;\!\!max} }=hf-\Phi \,\!}
Compton Effect
Change in wavelength of photons from an X-Ray source depends only
on scattering angle.
Δ
λ
=
h
2
m
(
1
−
cos
θ
)
{\displaystyle \Delta \lambda ={\frac {h}{2m}}\left(1-\cos \theta \right)\,\!}
Moseley's Law
Frequency of most intense X-Ray Spectrum (K-α) line for an element,
Atomic Number Z .
f
=
c
λ
=
M
K
α
(
Z
−
1
)
2
{\displaystyle f={\frac {c}{\lambda }}=M_{K_{\alpha }}(Z-1)^{2}}
M
K
α
=
2.47
×
10
15
{\displaystyle M_{K_{\alpha }}=2.47\times 10^{15}}
Planck's Radiation Law
I is spectral radiance (W m-2 sr-1 Hz-1 for frequency or W m-3 sr-1 for
wavelength), not simply intensity (W m-2 ).
I
(
ν
,
T
)
=
2
h
ν
3
c
2
1
e
h
ν
k
T
−
1
{\displaystyle I(\nu ,T)={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {h\nu }{kT}}-1}}\,\!}
I
(
λ
,
T
)
=
2
h
c
2
λ
5
1
e
h
c
λ
k
T
−
1
{\displaystyle I(\lambda ,T)={\frac {2hc^{2}}{\lambda ^{5}}}{\frac {1}{e^{\frac {hc}{\lambda kT}}-1}}}
The Assumptions of Quantum Mechanics
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1: State of a system
A system is completely specified at any one time by a Hilbert space vector.
2: Observables of a system
A measurable quantity corresponds to an operator with eigenvectors spanning the space .
3: Observation of a system
Measuring a system applies the observable's operator to the system and the system collapses into the observed eigenvector.
4: Probabilistic result of measurement
The probability of observing an eigenvector is derived from the square of its wavefunction .
5: Time evolution of a system
The way the wavefunction evolves over time is determined by Shrodinger's equation .
Quantum numbers are occur in the description of quantum states. There is one related to quantized atomic energy levels, and three related to quantized angular momentum.
Name
Symbol
Orbital Nomenclature
Values
Principal
n
shell
1
≤
n
,
n
∈
N
{\displaystyle 1\leq n,n\in \mathbf {N} \,\!}
Azimuth
Angular Momentum
l
subshell;
s orbital is listed as 0,
p orbital as 1,
d orbital as 2,
f orbital as 3,
etc for higher orbitals
0
≤
ℓ
≤
n
−
1
,
ℓ
∈
N
{\displaystyle 0\leq \ell \leq n-1,\ell \in \mathbf {N} \ }
Magnetic
Projection of Angular
Momentum
m l energy shift (orientation of
the subshell's shape)
−
ℓ
≤
m
ℓ
≤
ℓ
,
m
ℓ
∈
N
{\displaystyle -\ell \leq m_{\ell }\leq \ell ,m_{\ell }\in \mathbf {N} \ }
Spin
Projection of Spin
Angular Momentum
m s spin of the electron:
-1/2 = counter-clockwise,
+1/2 = clockwise
−
1
2
,
1
2
{\displaystyle -{\begin{matrix}{\frac {1}{2}}\end{matrix}},{\begin{matrix}{\frac {1}{2}}\end{matrix}}\ }
Quantum Wave-Function and Probability
edit
Born Interpretation of the Particle Wavefunction
Wavefunctions are probability distributions describing the space-time behaviour of a particle, distributed through space-time like a wave. It is the wave-particle duality characteristics incorperated into a mathematical function. This interpretation was due to Max Born.
Quantum Probability
Probability current (or flux) is a concept ; the flow of probability density .
The probability density is analogous to a fluid; the probability current is analogous to the fluid flow rate. In each case current is the product of density times velocity.
Usually the wave-function is dimensionless, but due to normalization integrals it may in general have dimensions of length to negative integer powers, since the integrals are with respect to space.
Operator (Common Name/s)
(Common) Symbol/s
Defining Equation
SI Unit
Dimension
(Quantum) Wave-function
ψ , Ψ
ψ
=
ψ
(
r
,
t
)
{\displaystyle \psi =\psi \left(\mathbf {r} ,t\right)\,\!}
Ψ
=
Ψ
(
r
,
t
)
{\displaystyle \Psi =\Psi \left(\mathbf {r} ,t\right)\,\!}
m-n
[L]-n
Wavefunction, Probability Density Function
ρ
ρ
(
r
,
t
)
=
|
Ψ
(
r
,
t
)
|
2
d
V
{\displaystyle \rho (\mathbf {r} ,t)=\left|\Psi (\mathbf {r} ,t)\right|^{2}\mathrm {d} V\,\!}
Probability Amplitude
A , N
Probability Current
Flow of Probability Density
J , I
Non-Relativistic
j
=
ℏ
2
m
i
(
Ψ
∗
∇
Ψ
−
Ψ
∇
Ψ
∗
)
{\displaystyle \mathbf {j} ={\frac {\hbar }{2mi}}\left(\Psi ^{*}\nabla \Psi -\Psi \nabla \Psi ^{*}\right)}
=
ℏ
m
I
m
(
Ψ
∗
∇
Ψ
)
{\displaystyle ={\frac {\hbar }{m}}\mathrm {Im} (\Psi ^{*}\nabla \Psi )}
=
R
e
(
Ψ
∗
ℏ
i
m
∇
Ψ
)
{\displaystyle =\mathrm {Re} \left(\Psi ^{*}{\frac {\hbar }{im}}\nabla \Psi \right)}
Properties and Requirements
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Normalization Integral
To be solved for probability amplitude.
R = Spatial Region Particle is definitley located in (including all space)
S = Boundary Surface of R .
∫
r
∈
R
|
Ψ
|
2
d
V
=
1
{\displaystyle \int _{\mathbf {r} \in R}\left|\Psi \right|^{2}\mathrm {d} V=1\,\!}
Law of Probability Conservation for Quantum Mechanics
∂
∂
t
∫
V
|
Ψ
|
2
d
V
+
∫
S
j
⋅
d
A
=
0
{\displaystyle {\frac {\partial }{\partial t}}\int _{V}|\Psi |^{2}\mathrm {d} V+\int _{S}\mathbf {j} \cdot \mathrm {d} \mathbf {A} =0}
Observable quntities are calculated by operators acting on the wave-function. The term potential alone often refers to the potential operator and the potential term in Schrödinger's Equation, but this is a misconception; rather the implied quantity is potential energy .
It is not immediatley obvious what the opeators mean in their general form, so component definitions are included in the table. Often for one-dimensional considerations of problems the component forms are useful, since they can be applied immediatley.
Operator (Common Name/s)
Component Definitions
General Definition
SI Unit
Dimension
Position
x
^
=
x
{\displaystyle {\hat {x}}=x\,\!}
y
^
=
y
{\displaystyle {\hat {y}}=y\,\!}
z
^
=
z
{\displaystyle {\hat {z}}=z\,\!}
r
^
=
r
{\displaystyle \mathbf {\hat {r}} =\mathbf {r} \,\!}
m
[L]
Momentum
p
^
x
=
−
i
ℏ
∂
∂
x
{\displaystyle {\hat {p}}_{x}=-i\hbar {\frac {\partial }{\partial x}}\,\!}
p
^
y
=
−
i
ℏ
∂
∂
y
{\displaystyle {\hat {p}}_{y}=-i\hbar {\frac {\partial }{\partial y}}\,\!}
p
^
z
=
−
i
ℏ
∂
∂
z
{\displaystyle {\hat {p}}_{z}=-i\hbar {\frac {\partial }{\partial z}}\,\!}
p
^
=
−
i
ℏ
∇
{\displaystyle \mathbf {\hat {p}} =-i\hbar \nabla \,\!}
J s m-1 = N s
[M] [L] [T]-1
Potential Energy
V
^
x
=
V
(
x
)
{\displaystyle {\hat {V}}_{x}=V(x)\,\!}
V
^
y
=
V
(
y
)
{\displaystyle {\hat {V}}_{y}=V(y)\,\!}
V
^
z
=
V
(
z
)
{\displaystyle {\hat {V}}_{z}=V(z)\,\!}
V
^
=
V
(
r
,
t
)
=
V
{\displaystyle {\hat {V}}=V\left(\mathbf {r} ,t\right)=V\,\!}
J
[M] [L]2 [T]-2
Energy
E
^
=
i
ℏ
∂
∂
t
{\displaystyle {\hat {E}}=i\hbar {\frac {\partial }{\partial t}}\,\!}
J
[M] [L]2 [T]-2
Hamiltonian
H
^
=
T
^
+
V
^
{\displaystyle {\hat {H}}={\hat {T}}+{\hat {V}}\,\!}
H
^
=
−
ℏ
2
2
m
∇
2
+
V
{\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V\,\!}
J
[M] [L]2 [T]-2
Angular Momentum
L
^
x
=
−
i
ℏ
(
y
∂
∂
z
−
z
∂
∂
y
)
{\displaystyle {\hat {L}}_{x}=-i\hbar \left(y{\partial \over \partial z}-z{\partial \over \partial y}\right)}
L
^
y
=
−
i
ℏ
(
z
∂
∂
x
−
x
∂
∂
z
)
{\displaystyle {\hat {L}}_{y}=-i\hbar \left(z{\partial \over \partial x}-x{\partial \over \partial z}\right)}
L
^
z
=
−
i
ℏ
(
x
∂
∂
y
−
y
∂
∂
x
)
{\displaystyle {\hat {L}}_{z}=-i\hbar \left(x{\partial \over \partial y}-y{\partial \over \partial x}\right)}
L
^
=
−
i
ℏ
r
×
∇
{\displaystyle \mathbf {\hat {L}} =-i\hbar \mathbf {r} \times \nabla }
J s = N s m-1
[M] [L]2 [T]-1
Spin Angular Momentum
S
^
x
=
ℏ
2
σ
x
{\displaystyle {\hat {S}}_{x}={\hbar \over 2}\sigma _{x}}
S
^
y
=
ℏ
2
σ
y
{\displaystyle {\hat {S}}_{y}={\hbar \over 2}\sigma _{y}}
S
^
z
=
ℏ
2
σ
z
{\displaystyle {\hat {S}}_{z}={\hbar \over 2}\sigma _{z}}
J s = N s m-1
[M] [L]2 [T]-1
Wavefunction Equations
edit
Schrödinger's Equation
H
^
Ψ
=
E
^
Ψ
{\displaystyle {\hat {H}}\Psi ={\hat {E}}\Psi }
1D
3D
Free Particle (V=0)
−
ℏ
2
2
m
d
2
d
x
2
Ψ
=
E
^
Ψ
{\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {\mathrm {d} ^{2}}{\mathrm {d} x^{2}}}\Psi ={\hat {E}}\Psi \,\!}
−
ℏ
2
2
m
∇
2
Ψ
=
E
^
Ψ
{\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\Psi ={\hat {E}}\Psi \,\!}
Time Independant
(
−
ℏ
2
2
m
d
2
d
x
2
+
V
)
Ψ
=
E
^
Ψ
{\displaystyle \left(-{\frac {\hbar ^{2}}{2m}}{\frac {\mathrm {d} ^{2}}{\mathrm {d} x^{2}}}+V\right)\Psi ={\hat {E}}\Psi \,\!}
(
−
ℏ
2
2
m
∇
2
+
V
)
Ψ
=
E
^
Ψ
{\displaystyle \left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V\right)\Psi ={\hat {E}}\Psi \,\!}
Time Dependant
(
−
ℏ
2
2
m
∂
2
∂
x
2
+
V
)
Ψ
=
i
ℏ
∂
∂
t
Ψ
{\displaystyle \left(-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}+V\right)\Psi =i\hbar {\frac {\partial }{\partial t}}\Psi \,\!}
(
−
ℏ
2
2
m
∇
2
+
V
)
Ψ
=
i
ℏ
∂
∂
t
Ψ
{\displaystyle \left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V\right)\Psi =i\hbar {\frac {\partial }{\partial t}}\Psi \,\!}
Dirac Equation
(
β
m
c
2
+
c
∑
k
=
1
3
α
k
p
k
)
Ψ
=
i
ℏ
∂
∂
t
Ψ
{\displaystyle \left({\boldsymbol {\beta }}mc^{2}+c\sum _{k=1}^{3}{\boldsymbol {\alpha }}_{k}p_{k}\right)\Psi =i\hbar {\frac {\partial }{\partial t}}\Psi }
β
{\displaystyle {\boldsymbol {\beta }}}
α
{\displaystyle {\boldsymbol {\alpha }}}
α
i
2
=
β
2
=
I
4
{\displaystyle {\boldsymbol {\alpha }}_{i}^{2}={\boldsymbol {\beta }}^{2}=\mathbf {I} _{4}}
α
i
α
j
+
α
j
α
i
=
0
{\displaystyle {\boldsymbol {\alpha }}_{i}{\boldsymbol {\alpha }}_{j}+{\boldsymbol {\alpha }}_{j}{\boldsymbol {\alpha }}_{i}=0\,}
α
i
β
+
β
α
i
=
0
{\displaystyle {\boldsymbol {\alpha }}_{i}{\boldsymbol {\beta }}+{\boldsymbol {\beta }}{\boldsymbol {\alpha }}_{i}=0\,}
Klien-Gorden Equation
[ 1]
(
−
1
c
2
∂
2
∂
t
2
+
∇
2
)
Ψ
=
(
m
0
c
ℏ
)
2
Ψ
{\displaystyle \left(-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}+\nabla ^{2}\right)\Psi =\left({\frac {m_{0}c}{\hbar }}\right)^{2}\Psi }
E
2
c
2
−
p
2
=
(
m
0
c
)
2
{\displaystyle {\frac {E^{2}}{c^{2}}}-p^{2}=\left(m_{0}c\right)^{2}}
Common Energies and Potential Energies
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The following energies are used in conjunction with Schrödinger's equation (and other variants). In fact the equation cannot be used for calculations unless the energies defined for it.
The concept of potential energy is important in analyzing probability amplitudes, since this energy confines particles to localized regions of space; the only exception to this is the free particle subject to zero potential energy.
V 0 = Constant Potential Energy
E 0 = Constant Total Energy
Potential Energy Type
Potential Energy V
Free Particle
0
One dimensional box
V
=
{
V
0
x
∈
[
a
,
b
]
0
x
∉
[
a
,
b
]
{\displaystyle V={\begin{cases}V_{0}&x\in [a,b]\\0&x\notin [a,b]\end{cases}}\,\!}
Harmonic Osscilator
V
=
1
2
k
x
2
{\displaystyle V={\frac {1}{2}}kx^{2}}
Electrostatic, Coulomb
V
=
q
1
q
2
4
π
ϵ
0
r
{\displaystyle V={\frac {q_{1}q_{2}}{4\pi \epsilon _{0}r}}}
Electric Dipole
V
=
−
p
⋅
E
{\displaystyle V=-\mathbf {p} \cdot \mathbf {E} }
Magnetic Dipole
V
=
−
m
⋅
B
{\displaystyle V=-\mathbf {m} \cdot \mathbf {B} }
Infinite Potential well
E
n
=
(
h
n
2
L
)
2
1
2
m
{\displaystyle E_{n}=\left({\frac {hn}{2L}}\right)^{2}{\frac {1}{2m}}\,\!}
n
∈
N
{\displaystyle n\in \mathbf {N} \,\!}
Wavefunction of a Trapped
Particle, One Dimensional Box
Ψ
n
(
x
)
=
A
sin
(
n
π
x
L
)
{\displaystyle \Psi _{n}(x)=A\sin \left({\frac {n\pi x}{L}}\right)\,\!}
n
∈
N
{\displaystyle n\in \mathbf {N} \,\!}
Hydrogen atom, orbital energy
E
n
=
−
m
e
4
8
ϵ
0
2
h
2
n
2
=
13.61
e
V
n
2
{\displaystyle E_{n}=-{\frac {me^{4}}{8\epsilon _{0}^{2}h^{2}n^{2}}}={\frac {13.61eV}{n^{2}}}\,\!}
n
∈
N
{\displaystyle n\in \mathbf {N} \,\!}
Quantum Numbers
spin projection quantum number
m
s
∈
{
−
1
/
2
,
+
1
/
2
}
{\displaystyle m_{s}\in \{-1/2,+1/2\}\,\!}
Orbital Electron Magnetic Dipole Moment
μ
o
r
b
=
−
e
L
/
2
m
{\displaystyle \mathbf {\mu } _{orb}=-e\mathbf {L} /2m\,\!}
Orbital Electron Magnetic Dipole Components
μ
o
r
b
,
z
=
−
m
L
μ
B
{\displaystyle \mathbf {\mu } _{orb,z}=-m_{\mathcal {L}}\mu _{B}\,\!}
Orbital, Spin, Electron Magnetic Dipole Moment
μ
s
=
−
e
S
/
m
=
g
q
S
/
2
m
{\displaystyle \mathbf {\mu _{s}} =-e\mathbf {S} /m=gq\mathbf {S} /2m\,\!}
Orbital Electron magnetic dipole moment
μ
o
r
b
=
−
e
L
o
r
b
/
2
m
{\displaystyle \mathbf {\mu } _{orb}=-e\mathbf {L} _{orb}/2m\,\!}
Orbital, Spin, Electron magnetic dipole moment Potential
U
=
−
μ
s
⋅
B
e
x
t
=
−
μ
s
,
z
B
e
x
t
{\displaystyle U=-\mathbf {\mu } _{s}\cdot \mathbf {B} _{ext}=-\mu _{s,z}B_{ext}\,\!}
Orbital, Electron Magnetic Dipole Moment Potential
U
=
−
μ
o
r
b
⋅
B
e
x
t
=
−
μ
o
r
b
,
z
B
e
x
t
{\displaystyle U=-\mathbf {\mu } _{orb}\cdot \mathbf {B} _{ext}=-\mu _{orb,z}B_{ext}\,\!}
Angular Momentum Components
L
z
=
m
L
ℏ
{\displaystyle L_{z}=m{\mathcal {L}}\hbar \,\!}
Spin Angular Momentum Magnitude
S
=
ℏ
s
(
s
+
1
)
{\displaystyle S=\hbar {\sqrt {s(s+1)}}\,\!}
Cutoff Wavelength
λ
m
i
n
=
h
c
/
K
0
{\displaystyle \lambda _{min}=hc/K_{0}\,\!}
Density of States
N
(
E
)
=
8
2
π
m
3
/
2
E
1
/
2
/
h
3
{\displaystyle N(E)=8{\sqrt {2}}\pi m^{3/2}E^{1/2}/h^{3}\,\!}
Occupancy Probability
P
(
E
)
=
1
/
(
e
(
E
−
E
F
)
/
k
T
+
1
)
{\displaystyle P(E)=1/(e^{(E-E_{F})/kT}+1)\,\!}
Hydrogen Atom Spectrum,
Rydberg Equation
1
λ
=
R
H
(
1
n
2
2
−
1
n
1
2
)
{\displaystyle {\frac {1}{\lambda }}=R_{H}\left({\frac {1}{n_{2}^{2}}}-{\frac {1}{n_{1}^{2}}}\right)\,\!}
Hydrogen Atom, radial probability density
P
(
r
)
=
4
r
2
a
3
e
2
r
/
a
{\displaystyle P(r)={\frac {4r^{2}}{a^{3}e^{2r/a}}}\,\!}
edit
↑ Particle Physics, B.R. Martin and G. Shaw, Manchester Physics Series 3rd Edition, 2009, ISBN 978-0-470-03294-7
![{\displaystyle V={\begin{cases}V_{0}&x\in [a,b]\\0&x\notin [a,b]\end{cases}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13db7be330fe5fe0b823eb4f90c2ddd19239704c)
