How is the Schrödinger equation written in the video?

Q: How is the Schrödinger equation written in the video?

The Schrödinger equation is written as $i\hbar\frac{\partial\Psi(x,t)}{\partial t}=-\frac{\hbar^2}{2m}\frac{\partial^2\Psi(x,t)}{\partial x^2}-U(x,t)\Psi(x,t)$. It features the imaginary unit $i$, the reduced Planck constant $\hbar$, the wavefunction $\Psi(x,t)$, mass $m$, and potential energy $U(x,t)$.

The Schrödinger equation is written as

i\hbar\frac{\partial\Psi(x,t)}{\partial t}=-\frac{\hbar^2}{2m}\frac{\partial^2\Psi(x,t)}{\partial x^2}-U(x,t)\Psi(x,t)

. It features the imaginary unit

i

, the reduced Planck constant

\hbar

, the wavefunction

\Psi(x,t)

, mass

m

, and potential energy

U(x,t)

Conditions

The equation is presented in the context of comparing wave equations.
It applies to the microscopic quantum realm.

Reasoning, step by step

Identify the left-hand side involving the first time derivative of the wavefunction.
Identify the right-hand side involving the second spatial derivative of the wavefunction.
Identify the potential energy term multiplying the wavefunction.
Note the constants: $i$ , $\hbar$ , and $m$ .

Example

The screen displays the full expression of the Schrödinger equation: $i\hbar\frac{\partial\Psi(x,t)}{\partial t}=-\frac{\hbar^2}{2m}\frac{\partial^2\Psi(x,t)}{\partial x^2}-U(x,t)\Psi(x,t)$ .

Common misconceptions

Confusing the wavefunction $\Psi$ with the classical displacement $f(x,t)$ .
Assuming the Schrödinger equation is a classical wave equation because it contains partial derivatives.

Watch the explanation

BilibiliEpisode 4: Wave equation

10:27 – 10:58Watch this moment ↗

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Answers are generated from source material and independently checked. Consult the original video or creator if something is unclear.