Physics Formula Quantum

Fundamental Quantum Physics Equations

Quantum physics is the branch of physics that deals with phenomena on a very small scale, such as atoms and subatomic particles. Unlike classical physics, quantum mechanics introduces concepts like wave-particle duality, quantization, and probability.

Key Quantum Physics Formulas

Planck's Equation

The energy of a quantum of electromagnetic radiation is given by:

\[ E = h f \]

Where:

\( E \) = Energy of the quantum (Joules)
\( h \) = Planck's constant (\( 6.626 \times 10^{-34} \) Js)
\( f \) = Frequency of radiation (Hz)

De Broglie Wavelength

Wave-particle duality is described by de Broglie's equation:

\[ \lambda = \frac{h}{p} \]

Where:

\( \lambda \) = Wavelength of a particle (m)
\( h \) = Planck's constant
\( p \) = Momentum of the particle (kg m/s)

Schrödinger Equation

The fundamental equation of quantum mechanics:

\[ i \hbar \frac{\partial}{\partial t} \Psi = \hat{H} \Psi \]

Where:

\( \Psi \) = Wave function
\( \hbar \) = Reduced Planck's constant (\( h/2\pi \))
\( \hat{H} \) = Hamiltonian operator

Heisenberg Uncertainty Principle

The uncertainty in position and momentum of a particle is given by:

\[ \Delta x \Delta p \geq \frac{\hbar}{2} \]

Where:

\( \Delta x \) = Uncertainty in position
\( \Delta p \) = Uncertainty in momentum

Quantum States and Probability

Quantum mechanics describes physical systems in terms of probability distributions rather than deterministic outcomes. The wave function \( \Psi \) represents the probability amplitude, and its squared modulus \( |\Psi|^2 \) gives the probability density.

Normalization Condition

The probability of finding a particle in all space must be 1:

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

Quantum Tunneling

Particles can penetrate potential barriers even when they lack sufficient energy, described by the transmission coefficient:

\[ T = e^{-2 \kappa a} \]

Where \( \kappa \) is the decay constant and \( a \) is the barrier width.

Examples

Example 1: Finding Energy of a Photon

Calculate the energy of a photon with a frequency of \( 5 \times 10^{14} \) Hz.

Using Planck’s equation:

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

Example 2: De Broglie Wavelength of an Electron

Find the wavelength of an electron moving with a momentum of \( 3.0 \times 10^{-24} \) kg m/s.

\[ \lambda = \frac{6.626 \times 10^{-34}}{3.0 \times 10^{-24}} = 2.2 \times 10^{-10} m \]

Example 3: Quantum Tunneling Probability

Consider an electron encountering a barrier of width \( 1 \times 10^{-9} \) m and height \( 5 eV \). The probability of tunneling is calculated using the transmission coefficient formula.

Applications of Quantum Physics

Quantum physics has numerous real-world applications, including:

Semiconductors and transistors
Quantum cryptography
Superconductivity
Quantum computing
Lasers and photonics

Conclusion

Quantum physics revolutionized our understanding of nature at the smallest scales. The formulas discussed provide a foundation for advanced studies in quantum mechanics and applications in modern technology.