$ \newcommand{\braket}[1]{\langle #1 \rangle} \newcommand{\abs}[2][]{\left\lvert#2\right\rvert_{\text{#1}}} \newcommand{\ket}[1]{\left\lvert#1 \right.\rangle} \newcommand{\bra}[1]{\langle\left. #1\right\rvert} \newcommand{\braket}[1]{\langle #1 \rangle} \newcommand{\dd}{\text{d}} \newcommand{\dv}[2]{\frac{\dd #1}{\dd #2}} \newcommand{\pdv}[2]{\frac{\partial}{\partial #1}} $
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$$\begin{align*} i\hbar\pdv{\Phi}{t}=-\frac{\hbar^2}{2m}\frac{\partial^2\Phi}{\partial x^2}+U(x)\Phi(x) \end{align*}$$

Significance

Just like $F=ma$ and classical laws of motion describes how classical systems evolve, the schrodinger's equation describes how a quantum mechanical system evolves. $$\begin{align*} i\hbar\frac{d}{dt}|\psi(t)\rangle=H(t)|\psi(t)\rangle \end{align*}$$

Derivation

The general wavefunction is $\Phi(x, t)=AE^{\frac{i}{\hbar}(p\hat{x}-Et)}$. Taking the derivative with respect to time gives us $$\begin{align*} \pdv{\Phi}{t}&=-\frac{i}{\hbar}E\Phi\\ \implies -\frac{\hbar}{i}\pdv{\Phi}{t}&=E\Phi \end{align*}$$ Assuming we have a free particle, then $E=\frac{\hat{p}^2}{2m}$, and $p$ writing in the position basis is $\frac{\hbar}{i}\pdv{}{x}$ To be continued

Associated Concepts

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$$\begin{align*} i\hbar\pdv{\Phi}{t}=-\frac{\hbar^2}{2m}\frac{\partial^2\Phi}{\partial x^2}+U(x)\Phi(x) \end{align*}$$

Significance

Just like $F=ma$ and classical laws of motion describes how classical systems evolve, the schrodinger's equation describes how a quantum mechanical system evolves. $$\begin{align*} i\hbar\frac{d}{dt}|\psi(t)\rangle=H(t)|\psi(t)\rangle \end{align*}$$

Derivation

The general wavefunction is $\Phi(x, t)=AE^{\frac{i}{\hbar}(p\hat{x}-Et)}$. Taking the derivative with respect to time gives us $$\begin{align*} \pdv{\Phi}{t}&=-\frac{i}{\hbar}E\Phi\\ \implies -\frac{\hbar}{i}\pdv{\Phi}{t}&=E\Phi \end{align*}$$ Assuming we have a free particle, then $E=\frac{\hat{p}^2}{2m}$, and $p$ writing in the position basis is $\frac{\hbar}{i}\pdv{}{x}$ To be continued

Associated Concepts

Coming soon
FullPage
overview
significance
derivation
related experiments
associated concepts
FullPage
overview
significance
derivation
related experiments
associated concepts
$$\begin{align*} i\hbar\pdv{\Phi}{t}=-\frac{\hbar^2}{2m}\frac{\partial^2\Phi}{\partial x^2}+U(x)\Phi(x) \end{align*}$$

Significance

Just like $F=ma$ and classical laws of motion describes how classical systems evolve, the schrodinger's equation describes how a quantum mechanical system evolves. $$\begin{align*} i\hbar\frac{d}{dt}|\psi(t)\rangle=H(t)|\psi(t)\rangle \end{align*}$$

Derivation

The general wavefunction is $\Phi(x, t)=AE^{\frac{i}{\hbar}(p\hat{x}-Et)}$. Taking the derivative with respect to time gives us $$\begin{align*} \pdv{\Phi}{t}&=-\frac{i}{\hbar}E\Phi\\ \implies -\frac{\hbar}{i}\pdv{\Phi}{t}&=E\Phi \end{align*}$$ Assuming we have a free particle, then $E=\frac{\hat{p}^2}{2m}$, and $p$ writing in the position basis is $\frac{\hbar}{i}\pdv{}{x}$ To be continued

Associated Concepts

Coming soon
$$\begin{align*} i\hbar\pdv{\Phi}{t}=-\frac{\hbar^2}{2m}\frac{\partial^2\Phi}{\partial x^2}+U(x)\Phi(x) \end{align*}$$

Significance

Just like $F=ma$ and classical laws of motion describes how classical systems evolve, the schrodinger's equation describes how a quantum mechanical system evolves. $$\begin{align*} i\hbar\frac{d}{dt}|\psi(t)\rangle=H(t)|\psi(t)\rangle \end{align*}$$

Derivation

The general wavefunction is $\Phi(x, t)=AE^{\frac{i}{\hbar}(p\hat{x}-Et)}$. Taking the derivative with respect to time gives us $$\begin{align*} \pdv{\Phi}{t}&=-\frac{i}{\hbar}E\Phi\\ \implies -\frac{\hbar}{i}\pdv{\Phi}{t}&=E\Phi \end{align*}$$ Assuming we have a free particle, then $E=\frac{\hat{p}^2}{2m}$, and $p$ writing in the position basis is $\frac{\hbar}{i}\pdv{}{x}$ To be continued

Associated Concepts

Coming soon
FullPage
overview
significance
derivation
related experiments
associated concepts