Atomic spectra

Time dependance

• classically, light is an electro magnetic wave with time behaviour

• time dependence of quantum states

  • A photon is a light quantum

    • lets describe its time dependence in the same way
    • the state of the electron in the atom is determined by energy
      • $|E_n(t)>=e^{-1E_{n}t/\hslash }|E_n(t=0)>$

  • for a system with only two energy levels

    • $|\psi ()>=\frac{1}{\sqrt{2}}|g>+\frac{1}{\sqrt{2}}|e>$

Schrodinger equation

• We know that from one state we use a matrix or operator to go to another state

  • goal is to find the operator that describes the time evolution

    • strategy: same as finding spin operator
      • write down expectation value
      • factor out $<\psi |\text{ and }|\psi >$

  • Expectation value for energy of two level atom

    • $<E>=P_s{E}_s+P_e+E_e$

• Hamiltonian

  • its the energy operator and its

    • $\left[\begin{array}{cc}{E}_g& 0\\ 0& E_e\end{array}\right]$ and to find the effect of the operator on the basis state we multiply it by the corresponding state

      • the Hamiltonian H doesnt change the energy states but multiplies it with the energy value

      • in linear algebra, such states are called eigenstates of the Hamiltonian operator

• The schrodinger equation

  • $i\hslash \frac{d}{dt}|\psi (t)>=H|\psi (t)>$
    • it describes how the system