Schrödinger equation — time-dependent, time-independent

WHY do we need an equation for $\Psi$ ?

HOW we build it (derivation from a free particle)

Why start here? Because this is the one quantum state whose energy and momentum we already know exactly. If the equation works here, we generalize.

Step 1 — extract $E$ by differentiating in time. Why? We want an operator that "pulls down" $E$ . $\frac{\partial \Psi}{\partial t}=-i\omega\,\Psi=-\frac{iE}{\hbar}\Psi \;\Rightarrow\; E\,\Psi=i\hbar\frac{\partial\Psi}{\partial t}.$

Step 2 — extract $p$ by differentiating in space. Why? We need $p^2$ to build kinetic energy. $\frac{\partial^2\Psi}{\partial x^2}=(ik)^2\Psi=-k^2\Psi=-\frac{p^2}{\hbar^2}\Psi \;\Rightarrow\; p^2\Psi=-\hbar^2\frac{\partial^2\Psi}{\partial x^2}.$

Step 3 — impose energy conservation. Why? This is the physics input. For a free particle $E=\frac{p^2}{2m}.$ Multiply by $\Psi$ and substitute the operators from Steps 1–2: $i\hbar\frac{\partial\Psi}{\partial t}=-\frac{\hbar^2}{2m}\frac{\partial^2\Psi}{\partial x^2}.$

Step 4 — add a potential. Why? A real particle feels forces. Total energy $E=\frac{p^2}{2m}+V(x)$ , so add $V(x)\Psi$ :

Figure — Schrödinger equation — time-dependent, time-independent

From TDSE to TISE — separation of variables

Substitute into the TDSE: $i\hbar\,\psi(x)\frac{d\phi}{dt}=\phi(t)\,\hat H\psi(x).$ Divide by $\psi\phi$ : $\underbrace{i\hbar\frac{1}{\phi}\frac{d\phi}{dt}}_{\text{time only}}=\underbrace{\frac{1}{\psi}\hat H\psi}_{\text{space only}}=E.$

Why a constant $E$ ? A function of $t$ alone equals a function of $x$ alone for all $x,t$ — only possible if both equal the same constant. That constant has units of energy and is the total energy $E$ .

The time part: $i\hbar\,\dot\phi=E\phi \Rightarrow \phi(t)=e^{-iEt/\hbar}$ .

The space part:

What the symbols mean (WHAT)

Worked Examples

Worked example 1 — Verify a plane wave solves the free TDSE

Take $\Psi=A e^{i(kx-\omega t)}$ , $V=0$ .

LHS: $i\hbar\,\partial_t\Psi=i\hbar(-i\omega)\Psi=\hbar\omega\,\Psi$ . Why? time-derivative pulls down $-i\omega$ .
RHS: $-\frac{\hbar^2}{2m}\partial_x^2\Psi=-\frac{\hbar^2}{2m}(-k^2)\Psi=\frac{\hbar^2k^2}{2m}\Psi$ . Why? two space derivatives give $(ik)^2=-k^2$ .
Equate: $\hbar\omega=\frac{\hbar^2k^2}{2m}$ , i.e. $E=\frac{p^2}{2m}$ . ✓ Why this matters? Confirms the equation reproduces the correct free-particle energy.

Worked example 2 — Infinite square well, ground state

$V=0$ for $0<x<L$ , infinite outside. Why infinite walls? They force $\psi=0$ at $x=0,L$ (the particle can't be in a region of infinite energy). Inside, TISE is $-\frac{\hbar^2}{2m}\psi''=E\psi \Rightarrow \psi''=-k^2\psi$ , $k=\sqrt{2mE}/\hbar$ . Solution $\psi=A\sin(kx)+B\cos(kx)$ . Boundary $\psi(0)=0\Rightarrow B=0$ . Why? $\cos 0=1\ne0$ . Boundary $\psi(L)=0\Rightarrow \sin(kL)=0\Rightarrow kL=n\pi$ . Why discard $n=0$ ? it gives $\psi\equiv0$ , no particle. Energies: $E_n=\frac{\hbar^2k^2}{2m}=\frac{n^2\pi^2\hbar^2}{2mL^2},\quad n=1,2,3\dots$ Why this matters? Quantization (discrete energies) emerges automatically from boundary conditions — not assumed.

Common Mistakes

Active Recall

Recall Self-test (hide answers)

What does $|\Psi|^2$ mean? → probability density.
Why first order in time? → so $\Psi(t_0)$ determines all future, deterministic evolution.
How do you get TISE from TDSE? → separation of variables, $V$ time-independent.
Why are energies quantized in a box? → boundary conditions $\psi(0)=\psi(L)=0$ .

Recall Feynman: explain to a 12-year-old

Imagine a foggy cloud that shows where a tiny ball might be. The thicker the fog, the more likely the ball is there. Schrödinger's equation is the weather rule that says how the fog drifts and changes as time passes. If you trap the ball in a box, the fog can only form certain neat wavy patterns — and each pattern comes with its own fixed energy. That's why a trapped quantum particle can only have special "menu" energies, not anything it likes.

Connections

de Broglie Hypothesis — supplies $p=\hbar k$ , the seed of the equation.
Wavefunction and Born Interpretation — meaning of $|\Psi|^2$ .
Particle in a Box — first application of TISE.
Quantum Harmonic Oscillator — TISE with $V=\tfrac12 m\omega^2x^2$ .
Hamiltonian Operator — the $\hat H$ in $\hat H\psi=E\psi$ .
Energy Quantization — emerges from boundary conditions.
Heisenberg Uncertainty Principle — consistent with wave nature of $\Psi$ .

What field describes a quantum particle's state?

The complex wavefunction

\Psi(x,t)

Write the time-dependent Schrödinger equation.

i\hbar\,\partial_t\Psi=-\frac{\hbar^2}{2m}\partial_x^2\Psi+V\Psi

Write the time-independent Schrödinger equation.

-\frac{\hbar^2}{2m}\psi''+V\psi=E\psi

, i.e.

\hat H\psi=E\psi

Why is the TDSE first order in time?

So that knowing

\Psi

now fully determines its future (deterministic evolution).

What operator gives energy from a plane wave?

i\hbar\,\partial_t

acting on

e^{-i\omega t}

returns

E

What operator relates to

p^2

-\hbar^2\partial_x^2

acting on

e^{ikx}

returns

p^2

How is TISE obtained from TDSE?

Separation of variables

\Psi=\psi(x)\phi(t)

when

V

is time-independent; both sides equal constant

E

What is the time factor of a stationary state?

\phi(t)=e^{-iEt/\hbar}

Why are stationary states "stationary"?

|\Psi|^2=|\psi|^2

is time-independent though

\Psi

has a rotating phase.

Energy levels of an infinite square well of width

L

E_n=\dfrac{n^2\pi^2\hbar^2}{2mL^2}

n=1,2,3,\dots

Normalization constant of

\psi=A\sin(\pi x/L)

[0,L]

A=\sqrt{2/L}

What does

|\Psi|^2\,dx

represent?

Probability of finding the particle in

[x,x+dx]

Normalization condition?

\int_{-\infty}^{\infty}|\Psi|^2dx=1

What is

\hat H

called and what does it represent?

The Hamiltonian; the total-energy operator.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, classical physics mein particle ki ek fixed position $x(t)$ hoti hai — Newton bata deta hai future. Quantum mein hum particle ko ek wavefunction $\Psi(x,t)$ se describe karte hain, jo complex hoti hai. Iska $|\Psi|^2$ batata hai ki particle kis jagah milne ki probability kitni hai. Schrödinger equation basically wahi rule hai jo batata hai ki ye $\Psi$ time ke saath kaise change hota hai — yaani quantum ka $F=ma$ .

Time-dependent form: $i\hbar\,\partial_t\Psi=\hat H\Psi$ . Ye time mein first order hai — matlab agar abhi ka $\Psi$ pata hai to poora future fix ho jata hai. Hum isko free particle ke de Broglie wave $e^{i(kx-\omega t)}$ se derive karte hain: time derivative se energy $E$ nikal aati hai, do baar space derivative se $p^2$ , aur fir energy conservation $E=p^2/2m+V$ daal do — bas, equation ban gaya.

Jab potential $V$ time pe depend nahi karta, hum trick lagate hain: $\Psi=\psi(x)\,\phi(t)$ split kar do (separation of variables). Time wala part $e^{-iEt/\hbar}$ ban jata hai, aur space wala part deta hai time-independent Schrödinger equation $\hat H\psi=E\psi$ — ye ek eigenvalue problem hai. Allowed energies $E$ hi eigenvalues hain. Yahin se quantization apne aap aata hai, jaise box mein particle ki $E_n=n^2\pi^2\hbar^2/2mL^2$ .

Important baat: "stationary state" ka matlab frozen nahi hai. $\Psi$ ka phase $e^{-iEt/\hbar}$ ghoomta rehta hai, par observable $|\Psi|^2$ constant rehta hai — isiliye "stationary" bolte hain. Exam mein yaad rakho: $|\Psi|^2$ hi probability density hai, $\Psi$ khud nahi, aur normalization $\int|\Psi|^2dx=1$ zaroor lagana.

Schrödinger equation — time-dependent, time-independent

WHY do we need an equation for $\Psi$ ?

HOW we build it (derivation from a free particle)

From TDSE to TISE — separation of variables

What the symbols mean (WHAT)

Worked Examples

Common Mistakes

Active Recall

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Modern Physics

Connections