Hamiltonian

Eigenvalue equation for total Hamiltonian

• Hermitian operators obey eigenvalue equations

  • $A|a_j>=a_j|a_j>$

• Example

  • find the eigen values and eigenstates of $S_y$

    • $S_y|\pm y>=\pm \lambda |\pm y>$
      • this we can take the $\det (S_y-\lambda I)$

The commutator

• A observable is said to commute if $[A,B]=0$

  • $[A,B]=AB-BA$
  • If they commute they have the same eigenstate, but different eigenvalue

• Theorem

  • Suppose A and B are herniation operators then $[A,B]=0$ if and only if there exists an orthonormal basis such that both A and B can be diagonal matrices in that basis (they are simultaneously diagonalizable)

Projection Operators

• Projection operators map a quantum state onto a specific subspace

  • We used them when deriving Spin matrices with calling them that

    • Recall
    • $<Sz\ge Pup\left(+\frac{\hslash }{2}\right)+Pdown\left(-\frac{\hslash }{2}\right)$
      • And we multiply the Bra Kets together because
        • When we multiply the Bra Kets together we get the projection operator
        • Pup is when up up Bra and Ket are multiplied
        • Pdown is when the Down bra And Ket are multiplied

The rules of Quantum mechanics

• we introduced the quantum state fo a system

• After preparation the system may evolve into another state or change

• Finally we chose to measure a physical observable of the quantum state

• For example an upward electron in a stern gerlach indicates spin value of hbar/2

• We find probabilities of the measurement outcomes by using the Born Rule

${\int }_a^{bx}{\sum }_i^u\psi (x)\partial xdx$