Flashcards
Quantum Mechanics I
Cards in this deck (19)
- 1Time Independent Schrodinger Equation\[-\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2}+V\psi=E\psi\]
\[\varphi(t)=e^{-iEt/\hbar}\] - 2Energy Eigenvalues of the Infinite Square Well of Width \(a\)\[E_n=\frac{n^2\pi^2\hbar^2}{2ma^2}\]
- 3Eigenfunctions of the Infinite Square Well of Width \(a\)\[\psi_n(x)=\sqrt{\frac{2}{a}}\sin{\Big(\frac{n\pi}{a}x\Big)}\]
- 4Energy Eigenvalues of a Simple Harmonic Oscillator\[E_n=(n+1/2)\hbar\omega\]
- 5Eigenfunctions of a Simple Harmonic Oscillator\[\psi_n(x)=\Big(\frac{m\omega}{\pi\hbar}\Big)^{1/4}(a_+)^ne^{-(m\omega/2\hbar)x^2}\]
- 6Energy Eigenvalues of a Delta-Function Potential \(V(x)=-\alpha\delta(x)\)\[E=-\frac{m\alpha^2}{2\hbar^2}\]
- 7Eigenfunctions of a Delta-Function Potential \(V(x)=-\alpha\delta(x)\)\[\psi(x)=\frac{\sqrt{m\alpha}}{\hbar}e^{-m\alpha|x|/\hbar^2}\]
- 8\[[S_i,S_j]\]\[i\sum_k\epsilon_{ijk}\hbar S_k\]
- 9\[[x_i,p_j]\]\[i\hbar\delta_{ij}\]
- 10\[L_i,L_j\]\[i\hbar\sum_k\epsilon_{ijk}L_k\]
- 11\[[J_i,J_j]\]\[i\hbar\epsilon_{ijk}J_k\]
- 12Heisenberg Equation of Motion\[\frac{dA_H(t)}{dt}=\frac{1}{i\hbar}[A_H,H]\]
\[[x_i,F(p)]=i\hbar\frac{\partial F}{\partial p_i}\]
\[[p_i,G(x)]=-i\hbar\frac{\partial G}{\partial x_i}\] - 13The Raising Operator\[a^{\dagger}=\sqrt{\frac{m\omega}{2\hbar}}\Big(x+\frac{ip}{m\omega}\Big)\]
\[a^{\dagger}|n>=\sqrt{n+1}|n+1>\] - 14The Lowering Operator\[a=\sqrt{\frac{m\omega}{2\hbar}}\Big(x-\frac{ip}{m\omega}\Big)\]
\[a|n>=\sqrt{n}|n-1>\] - 15\[J^2|j,m>\] \[J_z|j,m>\]\[j(j+1)\hbar^2|j,m>\]
\[m\hbar|j,m>\] - 16Spin Operator in Terms of the Pauli Operators\[S=\frac{\hbar}{2}\sigma\]
- 17Pauli Matrices
- 18Orthogonality of Spherical Harmonics\[\int_0^{2\pi}d\phi\int_{-1}^1 d(\cos{\theta})Y_{l'}^{m'^*}(\theta,\phi)Y_{l}^{m}(\theta,\phi)=\delta_{ll'}\delta_{mm'}\]
- 19\[\boldsymbol{S}^2\]\[\boldsymbol{S_1}^2+\boldsymbol{S_2}^2+2\boldsymbol{S_1}\cdot\boldsymbol{S_2}\]