Schr ö dinger’s Equation

1. Schrödinger’s Equation

2. Class Objectives • How do we get at the information in the wave function? • Introduce Schrödinger's equation. • Develop the time independent Schrödinger's Equation (TISE).

3. Schrödinger’s Equation • The fundamental problem in QM is:

4. Schrödinger’s Equation • The fundamental problem in QM is: given the wave function at some instant t=0, find the wave function at a subsequent time.

5. Schrödinger’s Equation • The fundamental problem in QM is: given the wave function at some instant t=0, find the wave function at a subsequent time. • The wave function gives the initial information.

6. Schrödinger’s Equation • The fundamental problem in QM is: given the wave function at some instant t=0, find the wave function at a subsequent time. • The wave function gives the initial information. • is determined from the Schrödinger equation.

7. Schrödinger’s Equation • In developing his theory, Schrödinger adopted de Broglie’s equations: , • As well he defined the total energy E as

8. Schrödinger’s Equation • For a particle acted on by a force F, must be found from Schrödinger's wave equation.

9. Schrödinger’s Equation • Schrödinger's wave equation for 1D is written as:

10. Schrödinger’s Equation • Schrödinger's wave equation for 1D is written as: • U(x) is the potential energy function for the force F. ie.

11. Schrödinger’s Equation • How do we obtain an equation for ?

12. Schrödinger’s Equation • How do we obtain an equation for ? • Schrödinger's equation is a partial differential equation for in terms of two variables.

13. Schrödinger’s Equation • How do we obtain an equation for ? • Schrödinger's equation is a partial differential equation for in terms of two variables. A standard technique is to look for solutions having separable form.

14. Schrödinger’s Equation • How do we obtain an equation for ? • Schrödinger's equation is a partial differential equation for in terms of two variables. A standard technique is to look for solutions having separable form. Ie.

15. Schrödinger’s Equation • How do we obtain an equation for ? • Schrödinger's equation is a partial differential equation for in terms of two variables. A standard technique is to look for solutions having separable form. Ie. • Where is a function of x only and a function of t only.

16. Schrödinger’s Equation • Substituting into Schrödinger's equation we get:

17. Schrödinger’s Equation • Substituting into Schrödinger's equation we get: • Dividing by gives

18. Schrödinger’s Equation • LHS is a function of x only.

19. Schrödinger’s Equation • LHS is a function of x only. • RHS is a function of t only.

20. Schrödinger’s Equation • LHS is a function of x only. • RHS is a function of t only. • Since changing t cannot effect LHS

21. Schrödinger’s Equation • LHS is a function of x only. • RHS is a function of t only. • Since changing t cannot effect LHS (changing x does not affect RHS)

22. Schrödinger’s Equation • LHS is a function of x only. • RHS is a function of t only. • Since changing t cannot effect LHS (changing x does not affect RHS), the differential can be separated into two ODEs.

23. Schrödinger’s Equation • Both sides must equal to the same separation constant.

24. Schrödinger’s Equation • Both sides must equal to the same separation constant. So that,

25. Schrödinger’s Equation • S1 is a 1st order ODE (Φ as a function of t). These have the solution

26. Schrödinger’s Equation • S1 is a 1st order ODE (Φ as a function of t). These have the solution

27. Schrödinger’s Equation • S1 is a 1st order ODE (Φ as a function of t). These have the solution • NB: You should verify this!

28. Schrödinger’s Equation • S1 is a 1st order ODE (Φ as a function of t). These have the solution • NB: You should verify this! • It is easy to show that C = E, the total energy.

29. Schrödinger’s Equation • S1 is a 1st order ODE (Φ as a function of t). These have the solution • NB: You should verify this! • It is easy to show that C = E, the total energy. • Thus

30. Schrödinger’s Equation • S1 is a 1st order ODE (Φ as a function of t). These have the solution • NB: You should verify this! • It is easy to show that C = E, the total energy. • Thus • And

31. Schrödinger’s Equation • is the time independent Schrödinger equation.

32. Schrödinger’s Equation • is the time independent Schrödinger equation. • We can write the solution to Schrödinger equation as

33. Schrödinger’s Equation • is the time independent Schrödinger equation. • We can write the solution to Schrödinger equation as • The expression gives a relationship between the time independent and dependent wave functions.

34. Schrödinger’s Equation • is the time independent Schrödinger equation. • We can write the solution to Schrödinger equation as • The expression gives a relationship between the time independent and dependent wave functions. • The solutions for are that of planes. ie

35. Schrödinger’s Equation • The functions of are called eigenfunctions.

36. Schrödinger’s Equation • The functions of are called eigenfunctions. • Solutions of Schrödinger’s equation are stationary states.

37. Schrödinger’s Equation • The functions of are called eigenfunctions. • Solutions of Schrödinger’s equation are stationary states. • This because they are time independent and the probability distributions are time independent.

38. Schrödinger’s Equation • The functions of are called eigenfunctions. • Solutions of Schrödinger’s equation are stationary states. • This because they are time independent and the probability distributions are time independent.

39. Schrödinger’s Equation • Because the probabilities are static they can be calculated from the time independent wave form.