Schrödinger equation
- Quantum Mechanics 1.7

in Physics on Quantum Mechanics views · 5 min
Schrödinger equation

from 「Modern Quantum Mechanics」: Sakurai, J. J.

The core of non-relativistic quantum mechanics.

Prerequisites

Like a photon, the momentum and energy of a quantum-like particle with wave-like properties and particle-like properties(i.e., duality) are summarized as follows. (Duality derived by De Broglie)

  • $ p=\dfrac{h}{\lambda}=\hbar k $
  • $ E=h\nu=\hbar\omega $

Since we are considering particles with this duality, let’s use wavefunction in classical mechanics.

  • $ \psi(\b{r},t)=Ae^{i(\b{k}\cdot\b{r}-\omega t)} $

Induction of time-dependent Schrödinger equation

Then,

  • $ \grad\psi = i\b{k}\psi = \dfrac{i}{\hbar}\b{p}\psi $
  • $ \b{p}\psi=-i\hbar\grad\psi $

Thus, the momentum operator in the position space is as follows.

\[ \hat{\b{p}}=-i\hbar\grad \]

We can write a Hamiltonian operator using this result:

  • $ \hat{H} = \hat{T}+\hat{V} = \dfrac{\hat{p}^2}{2m}+\hat{V} = -\dfrac{\hbar^2}{2m}\laplacian+\hat{V} $

Summarizing,

\[ \hat{H} = -\dfrac{\hbar^2}{2m}\laplacian + \hat{V} \]

Also,

  • $ \dps \pdv{\psi}{t} = -i\omega\psi = -\frac{i}{\hbar}E\psi $
  • $ \dps E\psi = i\hbar\pdv{ }{t}\psi $

Since the eigenvalue of a Hamiltonian operator is energy, we finally get time-dependent Schrödinger equation. Let’s rewrite the wavefunction $\psi$ to the state vector $\ket{\Psi(t)}$.

\[ i\hbar\pdv{ }{t}\ket{\Psi(t)} = \hat{H}\ket{\Psi(t)} \]

Induction of time-independent Schrödinger equation

Now let’s separate the state vector into terms for time and terms for Hamiltonian’s eigenfunctions assuming that the Hamiltonian is independent for time.

  • $ \ket{\Psi(t)}=T(t)\ket{\psi} $

Then we get time-independent Schrödinger equation. \[ \hat{H}\ket{\psi}=E\ket{\psi} \]

The solutions (eigenfunctions of the Hamiltonian) obtained through the separation of variables are not all solutions of the Schrödinger equation, but all solutions can be expressed by the basis containing those eigenfunctions.

\[ \begin{align*} i\hbar\pdv{ }{t}\ket{\Psi(t)} &= i\hbar\pdv{T}{t}\ket{\psi} \nl = \hat{H}\ket{\Psi(t)} &= ET\ket{\psi} \end{align*} \]

\[ \Rightarrow i\hbar\pdv{T}{t}=ET \]

\[ T(t) = e^{-i\frac{E}{\hbar}t} \] $T$ is called a time-evolution operator or a propagator.

\[ \therefore \ket{\Psi(t)}=e^{-i\frac{E}{\hbar}t}\ket{\psi} \]

Therefore, if we fully know the state vector at $t=0$, it is possible to describe how the system changes over time. \[ \therefore \ket{\Psi(t)}=\sum_n \brkt{\psi_n}{\Psi(0)}e^{-i\frac{E_n}{\hbar}t}\ket{\psi_n} \]