List of Quantum Physics Formulas Equations Latex Code

rockingdingo 2023-04-06 #physics #quantum

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 SchrÃ¶dinger 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 SchrÃ¶dinger 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. Planckâ??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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- Physics Formulary
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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.

The Uncertainty Principle

Equation

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

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

The SchrÃ¶dinger Equation

Equation

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

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 SchrÃ¶dinger 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

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

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 $Au_0=0$ holds

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 $L^2(L_\pm Y_{lm})=l(l+1)\hbar^2(L_\pm Y_{lm})$

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 schrÃ¶dinger equation of Spin
$V(r)$ : the potential operator for two particles with spin +- 1/2 h

The Dirac Formalism

Equation

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

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
$\{\gamma_\lambda,\gamma_\mu\}= \gamma_\lambda\gamma_\mu+\gamma_\mu\gamma_\lambda=2\delta_{\lambda\mu}$ : the dirac matrices
$\left(\gamma_\lambda\frac{\partial }{\partial x_\lambda}+\frac{m_0c}{\hbar}\right) \psi(\vec{x},t)=0$ : the Dirac equation
$\psi(x)=(\psi_1(x),\psi_2(x),\psi_3(x),\psi_4(x))$ : spinor

Navigation

1. Quantum Physics

Black Body Radiation

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The Compton Effect

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Quantum Wave Functions

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Operators in Quantum Physics

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The Uncertainty Principle

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The SchrÃ¶dinger Equation

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Parity

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The Tunnel Effect

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The Harmonic Oscillator

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The Angular Momentum Quantum

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Spin

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The Dirac Formalism

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