List of Quantum Physics Formulas Equations Latex Code

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In this blog, we will introduce most popuplar formulas in Quantum Physics. We will also provide latex code of the equations. Topics of Quantum Physics include Black Body Radiation, The Compton Effect, Quantum Wave Functions, Operators in Quantum Physics, The Uncertainty Principle, The Schrödinger Equation, Parity, The Tunnel Effect, Harmonic Oscillator, Angular momentum, Spin, the Dirac Formalism, and so on.

• 1. Quantum Physics
• Black Body Radiation
• The Compton Effect
• Quantum Wave Functions
• Operators in Quantum Physics
• The Uncertainty Principle
• The Schrödinger Equation
• Parity
• The Tunnel Effect
• Harmonic Oscillator
• Angular momentum
• Spin
• The Dirac Formalism

1. Quantum Physics

• Black Body Radiation

  Equation

  $$ w(f)=\frac{8\pi hf^3}{c^3}\frac{1}{{\rm e}^{hf/kT}-1} $$
  $$ w(\lambda)=\frac{8\pi hc}{\lambda^5}\frac{1}{{\rm e}^{hc/\lambda kT}-1} $$
  $$ P=A\sigma T^4 $$
  $$ T\lambda_{\rm max}=k_{\rm W} $$
  

  Latex Code

              w(f)=\frac{8\pi hf^3}{c^3}\frac{1}{{\rm e}^{hf/kT}-1} \\ 
              w(\lambda)=\frac{8\pi hc}{\lambda^5}\frac{1}{{\rm e}^{hc/\lambda kT}-1} \\
              P=A\sigma T^4 \\
              T\lambda_{\rm max}=k_{\rm W}
          

  Explanation

  Latex code for the Black Body Radiation. Planck’s law for the energy distribution for the radiation of a black body is listed above. I will briefly introduce the notations in this formulation.

  • $$P=A\sigma T^4$$: Stefan-Boltzmann's law for the total power density
  • $$T\lambda_{\rm max}=k_{\rm W}$$: Wien's law for the maximum
  
  

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• The Compton Effect

  Equation

  $$ \lambda'=\lambda+\frac{h}{mc}(1-\cos\theta)=\lambda+\lambda_{\rm C}(1-\cos\theta) $$
  

  Latex Code

              \lambda'=\lambda+\frac{h}{mc}(1-\cos\theta)=\lambda+\lambda_{\rm C}(1-\cos\theta)
          

  Explanation

  Latex code for The Compton Effect. If light is considered to consist of particles, the wavelength of scattered light can be derived as above. I will briefly introduce the notations in this formulation.

  
  

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• Quantum Wave Functions

  Equation

  $$ \Phi(k,t)=\frac{1}{\sqrt{h}}\int\Psi(x,t){\rm e}^{-ikx}dx $$
  $$ \Psi(x,t)=\frac{1}{\sqrt{h}}\int\Phi(k,t){\rm e}^{ikx}dk $$
  $$ v_{\rm g}=p/m $$
  $$ E=\hbar\omega $$
  $$ \left\langle f(t) \right\rangle=\int\hspace{-1.5ex}\int\hspace{-1.5ex}\int\Psi^* f\Psi d^3V $$
  $$ \left\langle f_p(t) \right\rangle=\int\hspace{-1.5ex}\int\hspace{-1.5ex}\int\Phi^*f\Phi d^3V_p $$
  $$ \left\langle f(t) \right\rangle=\left\langle \Phi|f|\Phi \right\rangle $$
  $$ \left\langle \Phi|\Phi \right\rangle=\left\langle \Psi|\Psi \right\rangle=1 $$
  

  Latex Code

              \Phi(k,t)=\frac{1}{\sqrt{h}}\int\Psi(x,t){\rm e}^{-ikx}dx \\ 
              \Psi(x,t)=\frac{1}{\sqrt{h}}\int\Phi(k,t){\rm e}^{ikx}dk \\
              v_{\rm g}=p/m \\
              E=\hbar\omega \\
              \left\langle f(t) \right\rangle=\int\hspace{-1.5ex}\int\hspace{-1.5ex}\int\Psi^* f\Psi d^3V \\ 
              \left\langle f_p(t) \right\rangle=\int\hspace{-1.5ex}\int\hspace{-1.5ex}\int\Phi^*f\Phi d^3V_p \\
              \left\langle f(t) \right\rangle=\left\langle \Phi|f|\Phi \right\rangle \\
              \left\langle \Phi|\Phi \right\rangle=\left\langle \Psi|\Psi \right\rangle=1
          

  Explanation

  Latex code for Quantum Wave Functions. If light is considered to consist of particles, the wavelength of scattered light can be derived as above. I will briefly introduce the notations in this formulation.

  • $$\psi$$: Wave Function
  • $$v_{\rm g}=p/m$$: Group Velocity
  • $$E=\hbar\omega$$: Energy
  • $$\left\langle f \right\rangle$$: Measure with the probability P of finding a particle somewhere. The expectation value $$\left\langle f \right\rangle$$ of a quantity f of a system.
  
  

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• Operators in Quantum Physics

  Equation

  $$ \int\psi_1^*A\psi_2d^3V=\int\psi_2(A\psi_1)^*d^3V $$
  $$ A\Psi=a\Psi $$
  $$ \Psi=\sum\limits_nc_nu_n $$
  $$ \frac{dA}{dt}=\frac{\partial A}{\partial t}+\frac{[A,H]}{i\hbar} $$
  

  Latex Code

              \int\psi_1^*A\psi_2d^3V=\int\psi_2(A\psi_1)^*d^3V \\
              A\Psi=a\Psi \\
              \Psi=\sum\limits_nc_nu_n \\
              \frac{dA}{dt}=\frac{\partial A}{\partial t}+\frac{[A,H]}{i\hbar}
          

  Explanation

  Latex code for Operators in Quantum Physics. I will briefly introduce the notations in this formulation.

  • $$ A\Psi=a\Psi $$: Eigenvalue Equation
  • $$ u_n $$: Eigenfunction
  • $$ \frac{dA}{dt}=\frac{\partial A}{\partial t}+\frac{[A,H]}{i\hbar} $$: The time-dependence of an operator is given by (Heisenberg)
  
  

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• The Uncertainty Principle

  Equation

  
  $$ (\Delta A)^2=\left\langle \psi|A_{\rm op}-\left\langle A \right\rangle|^2\psi \right\rangle=\left\langle A^2 \right\rangle-\left\langle A \right\rangle^2 $$
  $$ \Delta A\cdot\Delta B\geq \frac{1}{2} |\left\langle \psi|[A,B]|\psi \right\rangle| $$
  $$ \Delta E\cdot\Delta t\geq\hbar $$
  $$ [x,p_x]=i\hbar $$
  $$ \Delta p_x\cdot\Delta x\geq \frac{1}{2} \hbar $$
  $$ \Delta L_x\cdot\Delta L_y\geq \frac{1}{2} \hbar L_z $$
  

  Latex Code

              (\Delta A)^2=\left\langle \psi|A_{\rm op}-\left\langle A \right\rangle|^2\psi \right\rangle=\left\langle A^2 \right\rangle-\left\langle A \right\rangle^2 \\
              \Delta A\cdot\Delta B\geq \frac{1}{2} |\left\langle \psi|[A,B]|\psi \right\rangle| \\
              \Delta E\cdot\Delta t\geq\hbar \\
              [x,p_x]=i\hbar \\
              \Delta p_x\cdot\Delta x\geq \frac{1}{2} \hbar \\
              \Delta L_x\cdot\Delta L_y\geq \frac{1}{2} \hbar L_z
          

  Explanation

  Latex code for The Uncertainty Principle. I will briefly introduce the notations in this formulation.

  • $$ \Delta A $$: Uncertainty in A
  
  

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• The Schrödinger Equation

  Equation

  
  $$ -\dfrac{\hbar^{2}}{2m}\bigtriangledown ^{2} \psi +U\psi=E\psi = i\hbar \dfrac{\partial \psi}{\partial t} $$
  $$ H=p^2/2m+U, H\psi=E\psi $$
  $$ \psi(x,t)=\left(\sum+\int dE\right)c(E)u_E(x)\exp\left(-\frac{iEt}{\hbar}\right) $$
  $$ \displaystyle J=\frac{\hbar}{2im}(\psi^*\nabla\psi-\psi\nabla\psi^*) $$
  $$ \displaystyle\frac{\partial P(x,t)}{\partial t}=-\nabla J(x,t) $$
  

  Latex Code

              -\dfrac{\hbar^{2}}{2m}\bigtriangledown ^{2} \psi +U\psi=E\psi = i\hbar \dfrac{\partial \psi}{\partial t} \\
              H=p^2/2m+U, H\psi=E\psi \\
              \psi(x,t)=\left(\sum+\int dE\right)c(E)u_E(x)\exp\left(-\frac{iEt}{\hbar}\right) \\
              \displaystyle J=\frac{\hbar}{2im}(\psi^*\nabla\psi-\psi\nabla\psi^*) \\
              \displaystyle\frac{\partial P(x,t)}{\partial t}=-\nabla J(x,t)
          

  Explanation

  Latex code for the Schrödinger Equation. I will briefly introduce the notations in this formulation.

  • $$ p_{\rm op}=-i\hbar\nabla $$: The momentum operator
  • $$ x_{\rm op}=i\hbar\nabla_p $$: The position operator
  • $$ E_{\rm op}=i\hbar\partial/\partial t $$: The energy operator
  • $$ m $$: Mass
  • $$ U $$: Potential Energy
  • $$ E $$: Total Energy
  • $$ J $$: The current density
  • $$ \displaystyle\frac{\partial P(x,t)}{\partial t}=-\nabla J(x,t) $$: Conservation law
  
  

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• Parity

  Equation

  
  $$ {\cal P}\psi(x)=\psi(-x) $$
  $$ \psi(x)= \underbrace{\frac{1}{2} (\psi(x)+\psi(-x))}_{\rm even:~\hbox{$\psi^+$}}+ \underbrace{\frac{1}{2} (\psi(x)-\psi(-x))}_{\rm odd:~\hbox{$\psi^-$}} $$
  $$ \psi^+= \frac{1}{2} (1+{\cal P})\psi(x,t) $$
  $$ \psi^-= \frac{1}{2} (1-{\cal P})\psi(x,t) $$
  

  Latex Code

              {\cal P}\psi(x)=\psi(-x) \\
              \psi(x)= \underbrace{\frac{1}{2} (\psi(x)+\psi(-x))}_{\rm even:~\hbox{$\psi^+$}}+ \underbrace{\frac{1}{2} (\psi(x)-\psi(-x))}_{\rm odd:~\hbox{$\psi^-$}} \\
              \psi^+= \frac{1}{2} (1+{\cal P})\psi(x,t) \\
              \psi^-= \frac{1}{2} (1-{\cal P})\psi(x,t)
          

  Explanation

  Latex code for the Parity Equation. If the wavefunction is split into even and odd functions, it can be expanded into eigenfunctions of P. I will briefly introduce the notations in this formulation.

  • $$ {\cal P}\psi(x) $$: parity operator
  • $$ \psi^+= \frac{1}{2} (1+{\cal P})\psi(x,t) $$: Even Function
  • $$ \psi^-= \frac{1}{2} (1-{\cal P})\psi(x,t) $$: Odd Function
  
  

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• The Tunnel Effect

  Equation

  
  $$ \psi(x)=a^{-1/2}\sin(kx) $$
  $$ E_n=n^2h^2/8a^2m $$
  $$ \psi_1=A{\rm e}^{ikx}+B{\rm e}^{-ikx} $$
  $$ \psi_2=C{\rm e}^{ik'x}+D{\rm e}^{-ik'x} $$
  $$ \psi_3=A'{\rm e}^{ikx} $$
  $$ k'^2=2m(W-W_0)/\hbar^2 $$
  $$ k^2=2mW $$
  $$ T=|A'|^2/|A|^2 $$
  

  Latex Code

              \psi(x)=a^{-1/2}\sin(kx) \\
              E_n=n^2h^2/8a^2m \\
              \psi_1=A{\rm e}^{ikx}+B{\rm e}^{-ikx}  \\ 
              \psi_2=C{\rm e}^{ik'x}+D{\rm e}^{-ik'x}  \\
              \psi_3=A'{\rm e}^{ikx} \\
              k'^2=2m(W-W_0)/\hbar^2 \
              k^2=2mW \\
              T=|A'|^2/|A|^2
          

  Explanation

  Latex code for the tunnel Effect. I will briefly introduce the notations in this formulation.

  • $$ \psi(x)=a^{-1/2}\sin(kx) $$: Wavefunction of a particle in an infinitely high potential well
  • $$ E_n=n^2h^2/8a^2m $$: The energy levels
  • $$ \psi_1,\psi_2,\psi_3 $$: If 1, 2 and 3 are the areas in front, within and behind the potential well
  
  

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• The Harmonic Oscillator

  Equation

  $$ H=\frac{p^2}{2m}+\frac{1}{2} m\omega^2 x^2= \frac{1}{2} \hbar\omega+\omega A^\dagger A $$
  $$ A=\sqrt{\mbox{$\frac{1}{2}$}m\omega}x+\frac{ip}{\sqrt{2m\omega}} $$
  $$ A^\dagger=\sqrt{\mbox{$\frac{1}{2}$}m\omega}x-\frac{ip}{\sqrt{2m\omega}} $$
  $$ HAu_E=(E-\hbar\omega)Au_E $$
  $$ u_n=\frac{1}{\sqrt{n!}}\left(\frac{A^\dagger}{\sqrt{\hbar}}\right)^nu_0 $$
  $$ u_0=\sqrt[4]{\frac{m\omega}{\pi\hbar}}\exp\left(-\frac{m\omega x^2}{2\hbar}\right) $$
  $$ E_n=( \frac{1}{2} +n)\hbar\omega $$
  

  Latex Code

              H=\frac{p^2}{2m}+\frac{1}{2} m\omega^2 x^2= \frac{1}{2} \hbar\omega+\omega A^\dagger A \\
              A=\sqrt{\mbox{$\frac{1}{2}$}m\omega}x+\frac{ip}{\sqrt{2m\omega}} \\
              A^\dagger=\sqrt{\mbox{$\frac{1}{2}$}m\omega}x-\frac{ip}{\sqrt{2m\omega}} \\
              HAu_E=(E-\hbar\omega)Au_E \\
              u_n=\frac{1}{\sqrt{n!}}\left(\frac{A^\dagger}{\sqrt{\hbar}}\right)^nu_0 \\
              u_0=\sqrt[4]{\frac{m\omega}{\pi\hbar}}\exp\left(-\frac{m\omega x^2}{2\hbar}\right) \\
              E_n=( \frac{1}{2} +n)\hbar\omega
          

  Explanation

  Latex code for the Harmonic Oscillator Quantum Equation. I will briefly introduce the notations in this formulation.

  • $$ H $$: Hamiltonian
  • $$ A $$: Raising ladder operator
  • $$ A^\dagger $$: Lowering ladder operator
  • $$ u_0 $$: Eigenfunction for holds
  
  

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• The Angular Momentum Quantum

  Equation

  $$ [L_z,L^2]=[L_z,H]=[L^2,H]=0 $$
  $$ L_z=-i\hbar\frac{\partial }{\partial \varphi}=-i\hbar\left(x\frac{\partial }{\partial y}-y\frac{\partial }{\partial x}\right) $$
  $$ L_\pm=\hbar{\rm e}^{\pm i\varphi}\left(\pm\frac{\partial }{\partial \theta}+i\cot(\theta)\frac{\partial }{\partial \varphi}\right) $$
  $$ L_z(L_+Y_{lm})=(m+1)\hbar(L_+Y_{lm}) $$
  $$ L_z(L_-Y_{lm})=(m-1)\hbar(L_-Y_{lm}) $$
  $$ L^2(L_\pm Y_{lm})=l(l+1)\hbar^2(L_\pm Y_{lm}) $$
  
  

  Latex Code

              [L_z,L^2]=[L_z,H]=[L^2,H]=0 \\
              L_z=-i\hbar\frac{\partial }{\partial \varphi}=-i\hbar\left(x\frac{\partial }{\partial y}-y\frac{\partial }{\partial x}\right) \\
              L_\pm=\hbar{\rm e}^{\pm i\varphi}\left(\pm\frac{\partial }{\partial \theta}+i\cot(\theta)\frac{\partial }{\partial \varphi}\right) \\
              L_z(L_+Y_{lm})=(m+1)\hbar(L_+Y_{lm}) \\
              L_z(L_-Y_{lm})=(m-1)\hbar(L_-Y_{lm}) \\
              L^2(L_\pm Y_{lm})=l(l+1)\hbar^2(L_\pm Y_{lm})
          

  Explanation

  Latex code for the Angular Momentum Quantum Equation. I will briefly introduce the notations in this formulation.

  • $$ L $$: Angular momentum operators
  • $$ [L_+,L_z]=-\hbar L_+ $$: follows $$ L_z(L_+Y_{lm})=(m+1)\hbar(L_+Y_{lm}) $$
  • $$ [L_-,L_z]=\hbar L_- $$: follows $$ L_z(L_-Y_{lm})=(m-1)\hbar(L_-Y_{lm}) $$
  • $$ [L^2,L_\pm]=0 $$: follows $$L^2(L_\pm Y_{lm})=l(l+1)\hbar^2(L_\pm Y_{lm}) $$
  
  

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• Spin

  Equation

  $$ \vec{\vec{S}}= \frac{1}{2} \hbar\vec{\vec{\sigma}} $$
  $$ \vec{\vec{\sigma}}_x=\left(\begin{array}{cc}0&1\\1&0\end{array}\right) $$
  $$ \vec{\vec{\sigma}}_y=\left(\begin{array}{cc}0&-i\\i&0\end{array}\right) $$
  $$ \vec{\vec{\sigma}}_z=\left(\begin{array}{cc}1&0\\0&-1\end{array}\right) $$
  $$ i\hbar\frac{\partial \chi(t)}{\partial t}=\frac{eg_S\hbar}{4m}\vec{\sigma}\cdot\vec{B}\chi(t) $$
  $$ \vec{\sigma}=(\vec{\vec{\sigma}}_x,\vec{\vec{\sigma}}_y,\vec{\vec{\sigma}}_z) $$
  $$ V(r)=V_1(r)+\frac{1}{\hbar^2}(\vec{S}_1\cdot\vec{S_2})V_2(r)= V_1(r)+ \frac{1}{2} V_2(r)[S(S+1)-\frac{3}{2} ] $$
  

  Latex Code

              \vec{\vec{S}}= \frac{1}{2} \hbar\vec{\vec{\sigma}} \\
              \vec{\vec{\sigma}}_x=\left(\begin{array}{cc}0&1\\1&0\end{array}\right) \\
              \vec{\vec{\sigma}}_y=\left(\begin{array}{cc}0&-i\\i&0\end{array}\right) \\
              \vec{\vec{\sigma}}_z=\left(\begin{array}{cc}1&0\\0&-1\end{array}\right) \\
              i\hbar\frac{\partial \chi(t)}{\partial t}=\frac{eg_S\hbar}{4m}\vec{\sigma}\cdot\vec{B}\chi(t) \\
              \vec{\sigma}=(\vec{\vec{\sigma}}_x,\vec{\vec{\sigma}}_y,\vec{\vec{\sigma}}_z) \\
              V(r)=V_1(r)+\frac{1}{\hbar^2}(\vec{S}_1\cdot\vec{S_2})V_2(r)= V_1(r)+ \frac{1}{2} V_2(r)[S(S+1)-\frac{3}{2} ]
          

  Explanation

  Latex code for the Spin Equation. I will briefly introduce the notations in this formulation.

  • $$ \vec{\vec{S}}= \frac{1}{2} \hbar\vec{\vec{\sigma}} $$: Spin OperatorS
  • $$ S_z $$: Spinors
  • $$ \chi_+=(1,0) $$ : the state with spin up $$S_z= \frac{1}{2} \hbar $$
  • $$ \chi_-=(0,1)$$ : the state with spin down $$ S_z=- \frac{1}{2} \hbar $$
  • $$ |\alpha_+|^2,|\alpha_-|^2 $$ : the chance to find spin up and spin down
  • $$ i\hbar\frac{\partial \chi(t)}{\partial t}=\frac{eg_S\hbar}{4m}\vec{\sigma}\cdot\vec{B}\chi(t) $$: the schrödinger equation of Spin
  • $$ V(r) $$: the potential operator for two particles with spin +- 1/2 h
  
  

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• The Dirac Formalism

  Equation

  $$ E^2=m_0^2c^4+p^2c^2 $$
  $$ \displaystyle \left(\nabla^2-\frac{1}{c^2}\frac{\partial^2 }{\partial t^2}-\frac{m_0^2c^2}{\hbar^2}\right)\psi(\vec{x},t)=0 $$
  $$ \{\gamma_\lambda,\gamma_\mu\}= \gamma_\lambda\gamma_\mu+\gamma_\mu\gamma_\lambda=2\delta_{\lambda\mu} $$
  $$ \nabla^2-\frac{1}{c^2}\frac{\partial^2 }{\partial t^2}-\frac{m_0^2c^2}{\hbar^2}= \left\{\gamma_\lambda\frac{\partial }{\partial x_\lambda}-\frac{m_0c}{\hbar}\right\} \left\{\gamma_\mu\frac{\partial }{\partial x_\mu}+\frac{m_0c}{\hbar}\right\} $$
  $$ \gamma_k=\left(\begin{array}{cc}0&-i\sigma_k\\i\sigma_k&0\end{array}\right) $$
  $$ \gamma_4=\left(\begin{array}{cc}I&0\\0&-I\end{array}\right) $$
  $$ \left(\gamma_\lambda\frac{\partial }{\partial x_\lambda}+\frac{m_0c}{\hbar}\right) \psi(\vec{x},t)=0 $$
  $$ \psi(x)=(\psi_1(x),\psi_2(x),\psi_3(x),\psi_4(x)) $$
  

  Latex Code

              E^2=m_0^2c^4+p^2c^2 \\
              \displaystyle \left(\nabla^2-\frac{1}{c^2}\frac{\partial^2 }{\partial t^2}-\frac{m_0^2c^2}{\hbar^2}\right)\psi(\vec{x},t)=0 \\
              \{\gamma_\lambda,\gamma_\mu\}= \gamma_\lambda\gamma_\mu+\gamma_\mu\gamma_\lambda=2\delta_{\lambda\mu} \\
              \nabla^2-\frac{1}{c^2}\frac{\partial^2 }{\partial t^2}-\frac{m_0^2c^2}{\hbar^2}= \left\{\gamma_\lambda\frac{\partial }{\partial x_\lambda}-\frac{m_0c}{\hbar}\right\} \left\{\gamma_\mu\frac{\partial }{\partial x_\mu}+\frac{m_0c}{\hbar}\right\} \\
              \gamma_k=\left(\begin{array}{cc}0&-i\sigma_k\\i\sigma_k&0\end{array}\right) \\
              \gamma_4=\left(\begin{array}{cc}I&0\\0&-I\end{array}\right) \\
              \left(\gamma_\lambda\frac{\partial }{\partial x_\lambda}+\frac{m_0c}{\hbar}\right) \psi(\vec{x},t)=0 \\
              \psi(x)=(\psi_1(x),\psi_2(x),\psi_3(x),\psi_4(x))
          

  Explanation

  Latex code for the Dirac Formalism. I will briefly introduce the notations in this formulation.

  • $$ E^2=m_0^2c^4+p^2c^2 $$: relativistic equation
  • $$ \displaystyle \left(\nabla^2-\frac{1}{c^2}\frac{\partial^2 }{\partial t^2}-\frac{m_0^2c^2}{\hbar^2}\right)\psi(\vec{x},t)=0 $$: Klein-Gordon equation
  • $$ https://latex.codecogs.com/gif.latex?\{\gamma_\lambda,\gamma_\mu\}= \gamma_\lambda\gamma_\mu+\gamma_\mu\gamma_\lambda=2\delta_{\lambda\mu} $$: the dirac matrices
  • $$ \left(\gamma_\lambda\frac{\partial }{\partial x_\lambda}+\frac{m_0c}{\hbar}\right) \psi(\vec{x},t)=0 $$: the Dirac equation
  • $$ https://latex.codecogs.com/gif.latex?\psi(x)=(\psi_1(x),\psi_2(x),\psi_3(x),\psi_4(x)) $$: spinor
  
  

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