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 Schrödinger Equation, Parity, The Tunnel Effect, Harmonic Oscillator, Angular momentum, Spin, the Dirac Formalism, and so on.

    1. Quantum Physics

  • Black Body Radiation

    Equation


    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.

    • : Stefan-Boltzmann's law for the total power density
    • : Wien's law for the maximum

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  • The Compton Effect

    Equation


    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


    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.

    • : Wave Function
    • : Group Velocity
    • : Energy
    • : Measure with the probability P of finding a particle somewhere. The expectation value
    • of a quantity f of a system.

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  • Operators in Quantum Physics

    Equation


    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.

    • : Eigenvalue Equation
    • : Eigenfunction
    • : The time-dependence of an operator is given by (Heisenberg)

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  • The Uncertainty Principle

    Equation


    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.

    • : Uncertainty in A

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  • The Schrödinger Equation

    Equation


    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.

    • : The momentum operator
    • : The position operator
    • : The energy operator
    • : Mass
    • : Potential Energy
    • : Total Energy
    • : The current density
    • : Conservation law

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  • Parity

    Equation


    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.

    • : parity operator
    • : Even Function
    • : Odd Function

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  • The Tunnel Effect

    Equation


    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.

    • : Wavefunction of a particle in an infinitely high potential well
    • : The energy levels
    • : If 1, 2 and 3 are the areas in front, within and behind the potential well

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  • The Harmonic Oscillator

    Equation


    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.

    • : Hamiltonian
    • : Raising ladder operator
    • : Lowering ladder operator
    • : Eigenfunction for holds

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  • The Angular Momentum Quantum

    Equation


    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.

    • : Angular momentum operators
    • : follows
    • : follows
    • : follows

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  • Spin

    Equation


    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.

    • : Spin OperatorS
    • : Spinors
    • : the state with spin up
    • : the state with spin down
    • : the chance to find spin up and spin down
    • : the schrödinger equation of Spin
    • : the potential operator for two particles with spin +- 1/2 h

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  • The Dirac Formalism

    Equation


    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.

    • : relativistic equation
    • : Klein-Gordon equation
    • : the dirac matrices
    • : the Dirac equation
    • : spinor

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