Also, I am highly interested in quantum mechanics. Quantum mechanics is basically the study of how particles interact with each other on small and large scales. The real definition is Non-relativistic time-independent Schrödinger equation Summarized below are the various forms the Hamiltonian takes, with the corresponding Schrödinger equations and forms of wavefunction solutions. Notice in the case of one spatial dimension, for one particle, the partial derivative reduces to an ordinary derivative. One particle N particles One dimension \hat{H} = \frac{\hat{p}^2}{2m} + V(x) = -\frac{\hbar^2}{2m}\frac{d^2}{d x^2} + V(x) \begin{align}\hat{H} &= \sum_{n=1}^{N}\frac{\hat{p}_n^2}{2m_n} + V(x_1,x_2,\cdots x_N) \\ & = -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\frac{\partial^2}{\partial x_n^2} + V(x_1,x_2,\cdots x_N) \end{align} where the position of particle n is xn. E\Psi = -\frac{\hbar^2}{2m}\frac{d^2}{d x^2}\Psi + V\Psi E\Psi = -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\frac{\partial^2}{\partial x_n^2}\Psi + V\Psi \, . \Psi(x,t)=\psi(x) e^{-iEt/\hbar} \, . There is a further restriction — the solution must not grow at infinity, so that it has either a finite L2-norm (if it is a bound state) or a slowly diverging norm (if it is part of a continuum):[1] \| \psi \|^2 = \int |\psi(x)|^2\, dx.\, \Psi = e^{-iEt/\hbar}\psi(x_1,x_2\cdots x_N) for non-interacting particles \Psi = e^{-i{E t/\hbar}}\prod_{n=1}^N\psi(x_n) \, , \quad V(x_1,x_2,\cdots x_N) = \sum_{n=1}^N V(x_n) \, . Three dimensions \begin{align}\hat{H} & = \frac{\hat{\mathbf{p}}\cdot\hat{\mathbf{p}}}{2m} + V(\mathbf{r}) \\ & = -\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r}) \end{align} where the position of the particle is r = (x, y, z). \begin{align} \hat{H} & = \sum_{n=1}^{N}\frac{\hat{\mathbf{p}}_n\cdot\hat{\mathbf{p}}_n}{2m_n} + V(\mathbf{r}_1,\mathbf{r}_2,\cdots\mathbf{r}_N) \\ & = -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\nabla_n^2 + V(\mathbf{r}_1,\mathbf{r}_2,\cdots\mathbf{r}_N) \end{align} where the position of particle n is r n = (xn, yn, zn), and the Laplacian for particle n using the corresponding position coordinates is \nabla_n^2=\frac{\partial^2}{{\partial x_n}^2} + \frac{\partial^2}{{\partial y_n}^2} + \frac{\partial^2}{{\partial z_n}^2} E\Psi = -\frac{\hbar^2}{2m}\nabla^2\Psi + V\Psi E\Psi = -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\nabla_n^2\Psi + V\Psi \Psi = \psi(\mathbf{r}) e^{-iEt/\hbar} \Psi = e^{-iEt/\hbar}\psi(\mathbf{r}_1,\mathbf{r}_2\cdots
Posted on: Tue, 04 Mar 2014 08:17:29 +0000
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