Review Three Pictures of Quantum Mechanics

1. ReviewThree Pictures of Quantum Mechanics Simple Case: Hamiltonian is independent of time • Schrödinger Picture: Operators are independent of time; state vectors depend on time. • Heisenberg Picture: Operators depend on time; state vectors are independent of time. • Interaction pictures: Intermediate view; both state vectors and operators depend on time. All pictures are equivalent. We will use each at different times

2. Schrödinger Picture Operators are independent of time. Time dependence is in wave fucntion Integrate with respect to time: Re-iterate:

3. Schrödinger Picture Continue to re-iterate. If series converges, then: Note: Definition of the exponential of an operator is the exponential’s series expansion of that operator.

4. Heisenberg Picture States are independent of time. Operators carry time dependence. Showing time independence of this definition:

5. Heisenberg and Schrödinger Operators Complicated when Heisenberg Operator: Equation of motion for Heisenberg Operators At t=0: since

6. Interaction Picture Let’s divide the Hamiltonian into two parts: Usually H0 is a soluble problem. What are the effects of H1? Define: Generally:

7. What Is Second Quantization?Review of Simple Harmonic Oscillator Schrödinger Equation:

8. Solution to Simple Harmonic Oscillator Assume: Then: and where Wave functions are normalizable Hermite series must terminate

9. Matrix Notation Let: Define inner product: Orthonormality of wave functions gives: Since Schrödinger equation is linear and the set of eigenfunctions is complete: Complete solution: where

10. Energy Quantization from Commutation Relationships or Either

11. Raising and Lowering Operators Combining red equations in another way: Thus, either or lowers the state by one Thus the operator: Define dimensionless lowering operator: raises the state by one Likewise the operator: Define dimensionless raising operator:

12. Number Operator With these definitions: Number operator:

13. Ground State Assume that there is a lowest state such that: All other states can be built from the ground state by repeated applications of the raising operator:

14. Heisenberg States Are stationary in time. Time development is in the operators:

15. Step to Second Quantization Consider the complete set of time independent SHO Heisenberg states : The relationship between one state and another is the addition or subtraction of an elemental excitation (exciton) represented by the creation operator (raising operator) a† and the destruction operator (lowering operator) a respectively. Each exciton is represented by an the operator a† and has its own equation of motion given by: Second quantization is the process of considering excitations of a system as individual particles with their own equations of motion.