Hamiltonian H and its operator

\[E = \frac{mv^2}{2}+ U(x) \]

p = mv, because we can not determine a speed of a particle. Velocity is Frame-Dependent, Momentum is More Universal.

\[\nabla \cdot \Psi_p(x, t) \]

\[\nabla \times \Psi_p(x, t) \]

\[\nabla \Psi_p(x, t) \]

the last one is vector field. So in the conservative force:

\[F=\frac{dp}{dt}=m\frac{dv}{dt}=-\nabla U \]

\[H = \frac{p^2}{2m} + U(x) \]

in particle, de Broglie Hypothesis:

\[p = \frac{h}{\lambda} = \frac{2\pi \hbar}{\lambda} =k \hbar \]

For a particle with a definite momentum p, we expect this wavefunction to be a perfect plane wave, oscillating in space. A general plane wave can be written as:

\[\Psi_p(x, t) = A e^{i(kx - \omega t)} \]

A is the amplitude.

k=\(\frac{2\pi}{\lambda}\)​ is the wavenumber.

ω is the angular frequency.

\[\frac{\partial}{\partial x} \Psi_p(x, t) = \frac{\partial}{\partial x} \left( A e^{i(kx - \omega t)} \right) = A \cdot i k \cdot e^{i(kx - \omega t)} = i k \ \Psi_p(x, t) = i \frac{p}{\hbar} \ \Psi_p(x, t) \]

\[-i \hbar \frac{\partial}{\partial x} \Psi_p(x, t) = p \ \Psi_p(x, t) \]

A operator is:

• A mathematical instruction that transforms functions
• \(\hat{A}ψ=aψ\) - The operator acting on special states gives a definite value
• Operators are not fundamental - they emerge from the sum over paths.

\[\hat{H} = \frac{\hat{p}^2}{2m} + U(x) = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + U(x) \]

\[\hat{H} = -\frac{\hbar^2}{2m} {\nabla}^2 + U(r) \]

\[\hat{H} \Psi(r) = [-\frac{\hbar^2}{2m} {\nabla}^2 + U(r)] \Psi(r) = E \Psi(r) \]

where \(r=(x_1​,x_2​,…,x_n​)\) is the n-dimensional position vector.