Lecture 3: The Time Dependent Schrödinger Equation The material in this lecture is not covered in Atkins. It is required to understand postulate 6 and 11.5 The informtion of a wavefunction Lecture on-line The Time Dependent Schrödinger Equation (PDF) The time Dependent Schroedinger Equation (HTML) The time dependent Schrödinger Equation (PowerPoint) Tutorials on-line The postulates of quantum mechanics (This is the writeup for Dry-lab-II ( This lecture coveres parts of postulate 6) Time Dependent Schrödinger Equation The Development of Classical Mechanics Experimental Background for Quantum mecahnics Early Development of Quantum mechanics Audio-visuals on-line review of the Schrödinger equation and the Born postulate (PDF) review of the Schrödinger equation and the Born postulate (HTML) review of Schrödinger equation and Born postulate (PowerPoint **, 1MB) Slides from the text book (From the CD included in Atkins,**)

It is important to note that the particle is not distributed over a large region as a charge cloud It is the probability patterns (wave function) used to describe the electron motion that behaves like waves and satisfies a wave equation

We have and

The R.H.S. does not depend on t if we now assume that V is time independent. Thus, the L.H.S. must also be independent of t

Thus : The L.H.S. does not depend on x so the R.H.S. must also be independent of x and equal to the same constant, E.

We can now solve for f(t) : Or : Now integrating from time t=0 to t=to on both sides affords:

Or:

This is the time-independent Schroedinger Equation for a particle moving in the time independent potential V(x) It is a postulate of Quantum Mechanics that E is the total energy of the system

The total wavefunction for a one-dimentional particle in a potential V(x) is given by

Thus we can write without loss of generality for a particle in a time-independent potential This wavefunction is time dependent and complex. Let us now look at the corresponding probability density

Thus, states describing systems with a time-independent potential V(x) have a time-independent (stationary) probability density.

We say that systems that can be described by wave functions of the type