Quantum Mechanics Intro

Quantum mechanics is a fundamental theory in physics that describes the behavior of matter and energy on microscopic scales, such as atoms and subatomic particles. Unlike classical mechanics, it relies on probability and wave functions rather than deterministic paths, introducing concepts like superposition and entanglement. This guide provides an introduction to quantum mechanics, covering all basic formulas (Planck’s equation, de Broglie relation, Schrödinger equation, Heisenberg uncertainty principle), detailed examples, and practical applications to illustrate its revolutionary impact on modern science and technology.

Planck’s Equation (with de Broglie, Schrödinger, Uncertainty)

Planck’s equation relates the energy of a photon to its frequency, marking the birth of quantum theory:

\[ E = h f \]

Where:

\( E \): Energy (J)
\( h = 6.626 \times 10^{-34} \, \text{J·s} \): Planck’s constant
\( f \): Frequency (Hz)

Related Formulas:

de Broglie Relation: \( \lambda = \frac{h}{p} \), where \( p = mv \) (momentum)
Schrödinger Equation (Time-Independent, 1D): \( -\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} + V\psi = E\psi \) (simplified, \( \hbar = \frac{h}{2\pi} \))
Heisenberg Uncertainty Principle: \( \Delta x \cdot \Delta p \geq \frac{\hbar}{2} \)

Examples

Example 1: Planck’s Equation

Energy of a photon with \( f = 5 \times 10^{14} \, \text{Hz} \):

\[ E = h f \] \[ = (6.626 \times 10^{-34}) (5 \times 10^{14}) \] \[ = 3.313 \times 10^{-19} \, \text{J} \]

Example 2: de Broglie Wavelength

Wavelength of an electron with mass \( m = 9.11 \times 10^{-31} \, \text{kg} \), velocity \( v = 1 \times 10^6 \, \text{m/s} \):

\[ p = mv \] \[ = (9.11 \times 10^{-31}) (1 \times 10^6) \] \[ = 9.11 \times 10^{-25} \, \text{kg·m/s} \] \[ \lambda = \frac{h}{p} \] \[ = \frac{6.626 \times 10^{-34}}{9.11 \times 10^{-25}} \] \[ \approx 7.27 \times 10^{-10} \, \text{m} \]

Example 3: Schrödinger Equation (Energy Estimate)

For a particle in a box (\( V = 0 \), \( m = 9.11 \times 10^{-31} \, \text{kg} \), box length \( L = 1 \times 10^{-9} \, \text{m} \), \( n = 1 \)):

\[ E = \frac{n^2 h^2}{8 m L^2} \] \[ \hbar = \frac{h}{2\pi} \] \[ = \frac{6.626 \times 10^{-34}}{2 \cdot 3.1416} \] \[ \approx 1.055 \times 10^{-34} \, \text{J·s} \] \[ E = \frac{(1)^2 (6.626 \times 10^{-34})^2}{8 (9.11 \times 10^{-31}) (1 \times 10^{-9})^2} \] \[ = \frac{(6.626 \times 10^{-34})^2}{8 \cdot 9.11 \times 10^{-31} \cdot 1 \times 10^{-18}} \] \[ \approx \frac{4.39 \times 10^{-67}}{7.288 \times 10^{-48}} \] \[ \approx 6.02 \times 10^{-20} \, \text{J} \]

Example 4: Heisenberg Uncertainty Principle

Minimum uncertainty in position if momentum uncertainty \( \Delta p = 1 \times 10^{-25} \, \text{kg·m/s} \):

\[ \Delta x \cdot \Delta p \geq \frac{\hbar}{2} \] \[ \Delta x \geq \frac{\hbar}{2 \Delta p} \] \[ = \frac{1.055 \times 10^{-34}}{2 \cdot 1 \times 10^{-25}} \] \[ \approx 5.275 \times 10^{-10} \, \text{m} \]

Applications

Quantum mechanics drives modern technology and scientific advancements. Below are examples with calculations:

Example 1: Semiconductor Energy Levels

Energy of an electron in a quantum well (\( L = 5 \times 10^{-9} \, \text{m} \), \( m = 9.11 \times 10^{-31} \, \text{kg} \), \( n = 2 \)):

\[ E = \frac{n^2 h^2}{8 m L^2} \] \[ = \frac{(2)^2 (6.626 \times 10^{-34})^2}{8 (9.11 \times 10^{-31}) (5 \times 10^{-9})^2} \] \[ = \frac{4 \cdot 4.39 \times 10^{-67}}{8 \cdot 9.11 \times 10^{-31} \cdot 2.5 \times 10^{-17}} \] \[ \approx \frac{1.756 \times 10^{-66}}{1.822 \times 10^{-46}} \] \[ \approx 9.64 \times 10^{-21} \, \text{J} \]

Example 2: Laser Photon Energy

Energy of a photon from a laser with \( f = 4.74 \times 10^{14} \, \text{Hz} \):

\[ E = h f \] \[ = (6.626 \times 10^{-34}) (4.74 \times 10^{14}) \] \[ \approx 3.14 \times 10^{-19} \, \text{J} \]

Example 3: Electron Diffraction

Wavelength of an electron with \( v = 2 \times 10^6 \, \text{m/s} \):

\[ p = mv \] \[ = (9.11 \times 10^{-31}) (2 \times 10^6) \] \[ = 1.822 \times 10^{-24} \, \text{kg·m/s} \] \[ \lambda = \frac{h}{p} \] \[ = \frac{6.626 \times 10^{-34}}{1.822 \times 10^{-24}} \] \[ \approx 3.64 \times 10^{-10} \, \text{m} \]

Example 4: Quantum Computing (Uncertainty)

Minimum position uncertainty if \( \Delta p = 5 \times 10^{-26} \, \text{kg·m/s} \):

\[ \Delta x \cdot \Delta p \geq \frac{\hbar}{2} \] \[ \Delta x \geq \frac{1.055 \times 10^{-34}}{2 \cdot 5 \times 10^{-26}} \] \[ \approx 1.055 \times 10^{-9} \, \text{m} \]

Example 5: Atomic Transition Energy

Energy of a photon with \( f = 6 \times 10^{14} \, \text{Hz} \):

\[ E = h f \] \[ = (6.626 \times 10^{-34}) (6 \times 10^{14}) \] \[ = 3.976 \times 10^{-19} \, \text{J} \]

Example 6: Particle in a Box (Higher State)

Energy for \( n = 3 \), \( L = 2 \times 10^{-9} \, \text{m} \):

\[ E = \frac{n^2 h^2}{8 m L^2} \] \[ = \frac{(3)^2 (6.626 \times 10^{-34})^2}{8 (9.11 \times 10^{-31}) (2 \times 10^{-9})^2} \] \[ = \frac{9 \cdot 4.39 \times 10^{-67}}{8 \cdot 9.11 \times 10^{-31} \cdot 4 \times 10^{-18}} \] \[ \approx \frac{3.951 \times 10^{-66}}{2.914 \times 10^{-47}} \] \[ \approx 1.36 \times 10^{-19} \, \text{J} \]