Quantum Mechanics in a Nutshell

incomplete

• Expectation value is given by

$$<x>=\sum _{i}pro{b}_i{x}_i=\int {\psi }^{\star }x\psi dx$$

• “Real” Heisenberg Uncertainty Principle

$$\Delta x\Delta p\ge \frac{\hslash }{2}$$

• Uncertainty is given by

$$\Delta A^2=<A^2>-<A{>}^2$$

• Hermitian Operators
  • Operators linked to observables
  • Eigenfunctions orthogonal
  • Form complete set

$$\int {\phi }^{\star }(A\phi )dx=\int (A\phi {)}^{\star }\phi dx$$

• Matrix representation

$$<A>=\int {\psi }^{\star }\hat{A}\psi dx=\left(\begin{array}{cccc}{c}_1^{\star }& c_2^{\star }& \dots & \end{array}\right)\left(\begin{array}{cc}{A}_{11}& A_{12},\dots \\ & \\ A_{21}& \dots \\ & \\ \dots & \dots \end{array}\right)\left(\begin{array}{c}{c}_1\\ c_2\\ c_3\\ \dots \end{array}\right)$$

• No DiffEq representation of spin
• Spin emerges from Dirac Equation(reviewed in IQM)
• Commutation Relations