Schrdingers Equation Class Objectives l l l How
Schrödinger’s Equation
Class Objectives l l l How do we get at the information in the wave function? Introduce Schrödinger's equation. Develop the time independent Schrödinger's Equation (TISE).
Schrödinger’s Equation l The fundamental problem in QM is:
Schrödinger’s Equation l The fundamental problem in QM is: given the wave function at some instant t=0, find the wave function at a subsequent time.
Schrödinger’s Equation l l 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.
Schrödinger’s Equation l l l 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.
Schrödinger’s Equation l In developing his theory, Schrödinger adopted de Broglie’s equations: , l As well he defined the total energy E as
Schrödinger’s Equation l For a particle acted on by a force F, must be found from Schrödinger's wave equation.
Schrödinger’s Equation l Schrödinger's wave equation for 1 D is written as:
Schrödinger’s Equation l Schrödinger's wave equation for 1 D is written as: l U(x) is the potential energy function for the force F. ie.
Schrödinger’s Equation l How do we obtain an equation for ?
Schrödinger’s Equation l l How do we obtain an equation for ? Schrödinger's equation is a partial differential equation for in terms of two variables.
Schrödinger’s Equation l l 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.
Schrödinger’s Equation l l 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.
Schrödinger’s Equation l l l 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.
Schrödinger’s Equation l Substituting into Schrödinger's equation we get:
Schrödinger’s Equation l Substituting into Schrödinger's equation we get: l Dividing by gives
Schrödinger’s Equation l LHS is a function of x only.
Schrödinger’s Equation l l LHS is a function of x only. RHS is a function of t only.
Schrödinger’s Equation l l l LHS is a function of x only. RHS is a function of t only. Since changing t cannot effect LHS
Schrödinger’s Equation l l l 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)
Schrödinger’s Equation l l l 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.
Schrödinger’s Equation l Both sides must equal to the same separation constant.
Schrödinger’s Equation l Both sides must equal to the same separation constant. So that,
Schrödinger’s Equation l S 1 is a 1 st order ODE (Φ as a function of t). These have the solution
Schrödinger’s Equation l S 1 is a 1 st order ODE (Φ as a function of t). These have the solution
Schrödinger’s Equation l l S 1 is a 1 st order ODE (Φ as a function of t). These have the solution NB: You should verify this!
Schrödinger’s Equation l l l S 1 is a 1 st 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.
Schrödinger’s Equation l l S 1 is a 1 st 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
Schrödinger’s Equation l l l S 1 is a 1 st 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
Schrödinger’s Equation l is the time independent Schrödinger equation.
Schrödinger’s Equation l l is the time independent Schrödinger equation. We can write the solution to Schrödinger equation as
Schrödinger’s Equation l l l 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.
Schrödinger’s Equation l l 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
Schrödinger’s Equation l The functions of eigenfunctions. are called
Schrödinger’s Equation l l The functions of are called eigenfunctions. Solutions of Schrödinger’s equation are stationary states.
Schrödinger’s Equation l l l 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.
Schrödinger’s Equation l l l 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.
Schrödinger’s Equation l Because the probabilities are static they can be calculated from the time independent wave form.
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