FAD1022 L45 — Introduction to Quantum Mechanics — Formula Sheet

A comprehensive formula sheet extracted from Lecture 45: Introduction to Quantum Mechanics (Photon Momentum, Compton Effect, de Broglie Waves, Heisenberg Uncertainty Principle, and the 1D Infinite Square Well).

Physical Constants

Symbol Value Description
$h$ $6.626 \times 10^{-34}$ J·s Planck's constant
$\hbar$ $1.0546 \times 10^{-34}$ J·s Reduced Planck constant ($\hbar = h/2\pi$)
$m_e$ $9.11 \times 10^{-31}$ kg Electron mass
$m_p$ $1.67 \times 10^{-27}$ kg Proton mass
$c$ $3.00 \times 10^{8}$ m/s Speed of light in vacuum
$1\text{ eV}$ $1.602 \times 10^{-19}$ J Electron-volt conversion

1. De Broglie Wavelength & Wave-Particle Duality

De Broglie Wavelength

$$\lambda = \frac{h}{p} = \frac{h}{mv}$$

  • $\lambda$ — de Broglie wavelength (m)
  • $h$ — Planck's constant (J·s)
  • $p$ — momentum of the particle (kg·m/s)
  • $m$ — mass of the particle (kg)
  • $v$ — velocity of the particle (m/s)

De Broglie Wavelength from Kinetic Energy

For non-relativistic particles, since $KE = \frac{p^2}{2m}$, the wavelength can also be written as:

$$\lambda = \frac{h}{\sqrt{2m \cdot KE}}$$

  • $KE$ — kinetic energy (J)

2. Wave Functions & Probability

Probability Density (Born Interpretation)

$$P(x,t),dx = |\Psi(x,t)|^2,dx = \Psi^*(x,t)\Psi(x,t),dx$$

  • $\Psi(x,t)$ — wave function (complex-valued)
  • $\Psi^*(x,t)$ — complex conjugate of $\Psi$
  • $P(x,t)$ — probability density (m$^{-1}$)
  • $|\Psi(x,t)|^2$ — probability per unit length of finding the particle at position $x$ at time $t$

Normalization Condition

$$\int_{-\infty}^{\infty} |\Psi(x,t)|^2,dx = 1$$

Ensures the total probability of finding the particle somewhere in all space equals unity (dimensionless).

Requirements for Valid Wave Functions

A physically admissible wave function must be:

  1. Single-valued — one value at each point in space
  2. Continuous — no jumps or discontinuities
  3. Finite — must not diverge to infinity
  4. Square-integrable — $\displaystyle \int_{-\infty}^{\infty} |\Psi|^2,dx$ is finite

3. Heisenberg Uncertainty Principle

Position–Momentum Uncertainty

$$\Delta x \cdot \Delta p \geq \frac{\hbar}{2}$$

  • $\Delta x$ — uncertainty in position (m)
  • $\Delta p$ — uncertainty in momentum (kg·m/s)
  • $\hbar$ — reduced Planck constant (J·s)

Energy–Time Uncertainty

$$\Delta E \cdot \Delta t \geq \frac{\hbar}{2}$$

  • $\Delta E$ — uncertainty in energy (J)
  • $\Delta t$ — uncertainty in time / lifetime of the state (s)

Practical Form (commonly used in calculations)

Some textbooks and the lecture slides use:

$$\Delta x \cdot \Delta p \geq \frac{h}{4\pi}$$

$$\Delta E \cdot \Delta t \geq \frac{h}{4\pi}$$

(Note: $h/4\pi = \hbar/2$)

Minimum Uncertainty Estimates

From the uncertainty principle, minimum momentum uncertainty for confinement $\Delta x$:

$$\Delta p \geq \frac{\hbar}{2\Delta x}$$

Corresponding minimum velocity uncertainty for a particle of mass $m$:

$$\Delta v \geq \frac{\Delta p}{m} \geq \frac{\hbar}{2m,\Delta x}$$

4. Schrödinger Equation

Time-Dependent Schrödinger Equation (TDSE)

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

  • $i$ — imaginary unit ($i^2 = -1$)
  • $\hbar$ — reduced Planck constant (J·s)
  • $m$ — particle mass (kg)
  • $\Psi(x,t)$ — time-dependent wave function
  • $V(x)$ — potential energy function (J)
  • $E$ — energy eigenvalue (J)

Separation of Variables (for time-independent potentials)

$$\Psi(x,t) = \psi(x) \cdot e^{-iEt/\hbar}$$

  • $\psi(x)$ — spatial part of the wave function
  • $e^{-iEt/\hbar}$ — time-dependent phase factor

Time-Independent Schrödinger Equation (TISE)

$$-\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x)$$

Hamiltonian Operator Form

$$\hat{H}\psi(x) = E\psi(x)$$

Where the Hamiltonian operator is:

$$\hat{H} = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x)$$

  • $\hat{H}$ — Hamiltonian operator (total energy operator)
  • $\psi(x)$ — eigenfunction (spatial wave function)
  • $E$ — energy eigenvalue (allowed energy, J)

Terms in the TISE

Term Expression Physical Meaning
Kinetic energy operator $\displaystyle -\frac{\hbar^2}{2m}\frac{d^2}{dx^2}$ Quantum kinetic energy from $p = \hbar k$
Potential energy $V(x)$ Depends on the physical system (J)
Energy eigenvalue $E$ Allowed total energy of the state (J)
Wave function $\psi(x)$ Spatial part describing quantum state

5. Particle in a 1D Infinite Square Well (1D Box)

Potential Definition

$$V(x) = \begin{cases} 0 & \text{for } 0 < x < L \ \infty & \text{for } x \leq 0 \text{ or } x \geq L \end{cases}$$

  • $L$ — length of the box (m)

Boundary Conditions

$$\psi(0) = 0, \quad \psi(L) = 0$$

The wave function must vanish at the infinitely high walls.

General Solution Inside the Box

For $0 < x < L$, where $V = 0$:

$$\psi(x) = A\sin(kx) + B\cos(kx)$$

With:

$$k = \frac{\sqrt{2mE}}{\hbar}$$

  • $k$ — wave number (m$^{-1}$)
  • $A, B$ — constants determined by boundary conditions

Quantization Condition

Applying boundary conditions forces:

$$k_n L = n\pi \quad \Rightarrow \quad k_n = \frac{n\pi}{L}, \qquad n = 1, 2, 3, \ldots$$

  • $n$ — quantum number (positive integers only; $n = 0$ is forbidden)

Energy Eigenvalues (Quantized Energy Levels)

$$E_n = \frac{\hbar^2 k_n^2}{2m} = \frac{\hbar^2 \pi^2 n^2}{2mL^2} = \frac{n^2 h^2}{8mL^2}$$

  • $E_n$ — energy of the $n$-th level (J)
  • $n = 1, 2, 3, \ldots$ — quantum number

Alternative Forms of Energy Quantization

$$E_n = \frac{n^2 h^2}{8mL^2} = \frac{n^2 \pi^2 \hbar^2}{2mL^2}$$

Zero-Point Energy (Ground State)

$$E_1 = \frac{h^2}{8mL^2} = \frac{\pi^2 \hbar^2}{2mL^2} > 0$$

The particle can never be at rest inside the box — a direct consequence of the uncertainty principle.

Normalized Wave Functions

$$\psi_n(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right), \qquad n = 1, 2, 3, \ldots$$

  • Normalization constant: $\displaystyle A = \sqrt{\frac{2}{L}}$
  • The wave function is zero at $x = 0$ and $x = L$

Probability Density

$$|\psi_n(x)|^2 = \frac{2}{L}\sin^2\left(\frac{n\pi x}{L}\right)$$

  • Probability of finding the particle between $x$ and $x + dx$: $|\psi_n(x)|^2 dx$

Nodes

Each state $\psi_n(x)$ has $(n - 1)$ nodes inside the box (points where $\psi_n = 0$, excluding the walls).

$n$ Nodes inside box Description
1 0 Half sine wave (ground state)
2 1 Full sine wave (first excited)
3 2 1.5 sine waves
4 3 2 full sine waves

Energy Level Spacing

$$\Delta E = E_{n+1} - E_n = \frac{(n+1)^2 h^2}{8mL^2} - \frac{n^2 h^2}{8mL^2} = \frac{(2n+1)h^2}{8mL^2}$$

Relative spacing:

$$\frac{E_{n+1} - E_n}{E_n} = \frac{(n+1)^2 - n^2}{n^2} = \frac{2n+1}{n^2} \approx \frac{2}{n} \quad \text{as } n \to \infty$$

As $n \to \infty$, the discrete levels approach a classical continuum (Bohr's Correspondence Principle).

6. Energy Transitions & Photon Emission

Energy of Emitted Photon

When a particle transitions from a higher energy level $n_i$ to a lower level $n_f$:

$$\Delta E = E_{n_i} - E_{n_f} = \frac{h^2}{8mL^2}\left(n_i^2 - n_f^2\right)$$

Wavelength of Emitted Photon

$$\lambda = \frac{hc}{\Delta E} = \frac{hc}{E_{n_i} - E_{n_f}}$$

  • $\lambda$ — wavelength of emitted photon (m)
  • $c$ — speed of light (m/s)
  • $\Delta E$ — energy difference between levels (J)

7. Quick-Reference: Key Equations

Equation Context
$\displaystyle \lambda = \frac{h}{p} = \frac{h}{mv}$ De Broglie wavelength
$\displaystyle \lambda = \frac{h}{\sqrt{2m \cdot KE}}$ De Broglie wavelength from KE
$\displaystyle P(x) = \Psi(x,t)
$\displaystyle \int_{-\infty}^{\infty} \Psi
$\displaystyle \Delta x \cdot \Delta p \geq \frac{\hbar}{2}$ Heisenberg uncertainty (position-momentum)
$\displaystyle \Delta E \cdot \Delta t \geq \frac{\hbar}{2}$ Heisenberg uncertainty (energy-time)
$\displaystyle i\hbar\frac{\partial\Psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2\Psi}{\partial x^2} + V\Psi$ Time-Dependent Schrödinger Equation
$\displaystyle \Psi(x,t) = \psi(x) e^{-iEt/\hbar}$ Separation of variables
$\displaystyle -\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V\psi = E\psi$ Time-Independent Schrödinger Equation (TISE)
$\displaystyle \hat{H}\psi = E\psi$ TISE in Hamiltonian form
$\displaystyle E_n = \frac{n^2 h^2}{8mL^2}$ Particle in 1D box: quantized energy
$\displaystyle \psi_n(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right)$ Particle in 1D box: normalized wave function
$\displaystyle \psi_n(x)
$\displaystyle \lambda = \frac{hc}{\Delta E}$ Photon wavelength from transition

8. Unit Conversions & Useful Relations

Energy Conversion

$$1\text{ eV} = 1.602 \times 10^{-19}\text{ J}$$

$$E\text{ (J)} = E\text{ (eV)} \times 1.602 \times 10^{-19}$$

$$E\text{ (eV)} = \frac{E\text{ (J)}}{1.602 \times 10^{-19}}$$

Kinetic Energy Relations

$$KE = \frac{1}{2}mv^2 = \frac{p^2}{2m}$$

$$p = mv = \sqrt{2m \cdot KE}$$

Reduced Planck Constant

$$\hbar = \frac{h}{2\pi} = 1.0546 \times 10^{-34}\text{ J·s}$$

Related