Coulomb's Law: Opposite electrical charges attract each other, while matching charges push apart.

Limits the simultaneous precision of measuring position (\(\Delta x\)) and momentum (\(\Delta p_x\)):

Limits the simultaneous precision of measuring position (\(\Delta x\)) and momentum (\(\Delta p_x\)):

Heisenberg uncertainty principle reversed

ave-Particle Duality & De Broglie Relations

• Planck–Einstein Relation: Connects the energy (\(E\)) of a photon to its frequency (\(\nu \)):

  
  

  \(E=h\nu =\hbar \omega \)

  
  

  (where \(h\) is Planck's constant, \(\hbar = \frac{h}{2\pi}\) is the reduced Planck constant, and \(\omega \) is angular frequency).

• De Broglie Wavelength: Relates the wavelength (\(\lambda \)) of matter waves to momentum (\(p\)):

  
  

  \(\lambda =\frac{h}{p}\implies p=\hbar k\)

  
  

  (where \(k\) is the wave number). [1, 2]

2. The Schrödinger Equations

• Time-Dependent Schrödinger Equation (TDSE): Governs how a quantum wave function (\(\Psi \)) evolves over time:

  
  

  \(i\hbar \frac{\partial }{\partial t}\Psi (\mathbf{r},t)=\^{H}\Psi (\mathbf{r},t)\)

• Time-Independent Schrödinger Equation (TISE): Used for stationary states with a time-independent potential:

  
  

  \(\^{H}\psi (\mathbf{r})=E\psi (\mathbf{r})\)

• Hamiltonian Operator (\(\^{H}\)): Represents the total energy (kinetic + potential energy) in 3D: [1, 2, 3, 4]

  
  

  \(\^{H}=-\frac{\hbar ^{2}}{2m}\nabla ^{2}+V(\mathbf{r})\)

3. Heisenberg Uncertainty Principle

• Position and Momentum: Limits the simultaneous precision of measuring position (\(\Delta x\)) and momentum (\(\Delta p_x\)):

  
  

  \(\Delta x\cdot \Delta p_{x}\ge \frac{\hbar }{2}\)

• Energy and Time: Limits the precision of energy (\(\Delta E\)) and time (\(\Delta t\)): [1]

  
  

  \(\Delta E\cdot \Delta t\ge \frac{\hbar }{2}\)

4. Quantum Operators & Commutation Relations

• Position Operator: \(\hat{x} = x\)

• Momentum Operator (1D): \(\hat{p}_x = -i\hbar \frac{\partial}{\partial x}\)

• Canonical Commutation Relation: Defines the fundamental incompatibility of measuring position and momentum simultaneously: [1, 2]

  
  

  \([\^{x},\^{p}_{x}]=\^{x}\^{p}_{x}-\^{p}_{x}\^{x}=i\hbar \)

5. Probability & Wave Function Normalization

• Born Rule (Probability Density): The probability of finding a particle at a given point is the square of the wave function's absolute value:

  
  

  \(P=|{}\Psi (\mathbf{r},t)|{}^{2}\)

• Normalization Condition: Ensures the total probability of finding the particle across all space equals 1: [1, 2, 3]

  
  

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

Would you like me to expand on a specific area, such as angular momentum operators, the hydrogen atom solutions, or spin matrices?

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  11:05

• Heisenberg uncertainty principle rev means you can know the position and momentum

• Quantum Operators & Commutation Relations

  • Position Operator: \(\hat{x} = x\)

  • Momentum Operator (1D): \(\hat{p}_x = -i\hbar \frac{\partial}{\partial x}\)

  • Canonical Commutation Relation: Defines the fundamental incompatibility of measuring position and momentum simultaneousl

all reversed, implications obvious.