CHAPTER 10 -- QUANTUM MECHANICS FUNDAMENTALS
Quantum mechanics describes the physics of the very small, where classical mechanics fails. It introduces probabilism and wave-particle duality.
10.1 QUANTIZATION
Energy and other physical quantities often exist in discrete "packets" or quanta rather than continuous ranges.
- Planck's Hypothesis: Energy of light is quantized. E = h * f.
- Bohr Atom: Angular momentum of electrons is quantized. L = n * h_bar.
10.2 WAVEFUNCTIONS
The state of a quantum system is fully described by a wavefunction, Psi(x, t).
- Psi is a complex-valued function.
- The Born Rule states that the probability density of finding a particle at position x is given by |Psi(x, t)|^2 = Psi * Psi_conjugate.
- The total probability of finding the particle somewhere in space must be 1 (Normalization).
10.3 OPERATORS
In quantum mechanics, physical observables (position, momentum, energy) are represented by mathematical operators acting on the wavefunction.
- Position operator: x_hat = x
- Momentum operator: p_hat = -i * h_bar * d/dx
- Energy operator (Hamiltonian): H_hat
10.4 ENERGY EIGENSTATES
The Schrodinger Equation describes how the wavefunction evolves.
Time-Dependent Schrodinger Equation:
i * h_bar * dPsi/dt = H_hat * Psi
Time-Independent Schrodinger Equation (for stationary states):
H_hat * psi(x) = E * psi(x)
Here, psi(x) is an eigenfunction (eigenstate) of the Hamiltonian, and E is the corresponding eigenvalue (energy level).
If a system is in an eigenstate, it has a definite energy.
10.5 SELECTION RULES
Transitions between energy levels occur via the absorption or emission of photons. However, not all transitions are allowed. "Selection rules" derived from conservation laws (angular momentum, parity) dictate which transitions are "allowed" (high probability) and which are "forbidden" (low probability).
Example: In electric dipole transitions, the change in orbital angular momentum quantum number (l) must be Delta_l = +/- 1.
