Time-Dependent Schrödinger Equation #
$$i\hslash\frac{d}{dt}|\Psi(t)\rangle=\hat{H}|\Psi(t)\rangle$$This is because
$$E=\frac{P^2}{2m}+V=H$$Potential energy \(V\)
That is,
For the 1-dimensional case:
$$j\hslash\frac{\partial\Psi}{\partial t}=-\frac{\hslash^2}{2m}\frac{\partial^2\Psi}{\partial x^2}+V\Psi$$For the 3-dimensional case:
$$i\hslash\frac{\partial}{\partial t}\Psi(x,t)=\Biggl[\frac{-\hslash^2}{2m}\nabla^2+V(x,t)\Biggr]\Psi(x,t)$$Hamiltonian H #
Analytical Mechanics (Classical Mechanics) #
$$H=H(q,p;t)=T+V$$- Kinetic energy \(T\)
- Potential energy \(V\)
- Generalized coordinates \(q\)
- Generalized momentum \(p\)
Time \(t\)
Quantum Mechanics #
While this is not the technical definition of the Hamiltonian in classical mechanics, it is the most commonly adopted form. Combining these yields the form used in the Schrödinger equation.
$$ \begin{equation} \begin{aligned} \hat{H} =\hat{T}+\hat{V} =\frac{\hat{P}\cdotp\hat{P}}{2m}+V(r,t) =-\frac{\hslash^2}{2m}\nabla^2+V(r,t) \end{aligned} \end{equation} $$Dirac Constant (Reduced Planck Constant) #
$$ \begin{equation} \begin{aligned} \hslash\equiv\frac{h}{2\pi} =1.054 571 817…\times 10^{-34}J\cdotp s =6.582119569…\times 10^{-16}eV\cdotp s \end{aligned} \end{equation} $$Physical Significance #
$$E=h\nu=\frac{h}{2\pi}\cdotp 2\pi\nu =\hslash\omega$$$$P=\frac{h}{\lambda}=\frac{h}{2\pi}\frac{2\pi}{\lambda} =\hslash k$$Laplacian #
$$\nabla^2=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}$$Time-Independent Schrödinger Wave Equation #
$$-\frac{\hslash^2}{2m}\frac{d^2\varphi (x)}{dx^2}+\lbrace V(x)-E\rbrace\varphi (x)=0$$- Energy eigenvalue \(E\)
- Eigenfunction \(\varphi (x)\)
\(\varphi(x)\) determines the amplitude of \(\Psi(x,t)\).
Probability Density #
$$P_r=\Psi^*\cdotp\Psi=|\Psi|^2$$A wave function that satisfies \(\intop_V\Psi^*\cdotp\Psi dxdydz=1\) is called a wave function normalized to 1.