1. Schrodinger wave equation
► The position of a particle is distributed through space like the amplitude of a wave.
► In quantum mechanics, a wavefunction describes the motion and location of a particle.
► A wavefunction is just a mathematical function which may be large in one region, small in others and zero elsewhere.

2. Concepts of Wave function
► If the a wavefunction is large at a particular point (i.e., the amplitude of the wave is large), then the particle has a high probability of being found at that point. If the wavefunction is zero at a point, then the particle will not be found there.
► The more rapidly a wavefunction changes from place to place (i.e., the greater the curvature of the wave), the higher kinetic energy of the particle it describes.

3. Schrodinger wave equation
for a 1-D system
for a 3-D system
where
for Cartesian coordinate
for spherical coordinate

4. System with Spherical Symmetry
where

5. Eigenstate ► in general, the Schrodinger eqn is
► H is the Hamiltonian operator, i.e., the energy operator.
► Schrodinger eqn is an eigenvalue equation
(operator)(function) = (constant) x (same function)
eigenvalue
eigenfunction

6. Operator ►(operator) (function) = (constant) x (same function)
The factor is called the eigenvalue of the operator
► The function (which must be the same on each side in an eigenvalue equation) is called an eigenfunction and is different for each eigenvalue.
► An eigenvalue is a measurable property of a system, an observable. Each observable has a corresponding operator.
(operator for observable) (wavefunction) = (value of observable) x (wavefunction)

7. Examples of Operators
► A basic postulate of quantum mechanics is the form of the operator for linear momentum.
► If we want to find the linear momentum of a particle in the x direction, we use the following eigenvalue equation.
► The eigenvalue, p, is the momentum.

8. Correspondence
► The kinetic energy operator is then created from the momentum operator.
► If we want to find the kinetic energy of a particle in the x direction, we use the following eigenvalue equation.
the eigenvalue, E, is the kinetic energy

9. What is the wavefunction?
► So what’s the big deal? This should be straight forward??
► Just carry out some operation on a wavefunction, divide the result by the same wavefunction and you get the observable you want.
► Each system has its own wavefunction (actually many wavefunctions) that need to be found before making use of an eigenvalue equation.
► The Schrodinger equation and Born’s interpretation of wavefunctions will guide us to the correct form of the wavefunction for a particular system.

10. Born interpretation of the wavefunction
► In the wave theory of light, the square of the amplitude of an electromagnetic wave in a particular region is interpreted as its intensity in that region.
► In quantum mechanics, the square of the wavefunction at some point is proportional to the probability of finding a particle at that point.

11. Interpretation probability density
no negative or complex probability densities
written
Both large positive and large negative regions of a wavefunction correspond to a high probability of finding a particle in those regions.

12. Normalization constant
► A normalization constant is found which will insure that the probability of finding a particle within all allowed space is 100%.

13. Born interpretation of the wavefunction
► There are several restrictions on the acceptability of wavefunctions.
► The wavefunction must not be infinite anywhere. It must be finite everywhere.
► The wavefunction must be single-valued. It can have only one value at each point in space.
► The 2nd derivative of the wavefunction must be well-defined everywhere if it is to be useful in the Schrodinger wave equation.

14. Born interpretation of the wavefunction
► The 2nd derivative of a function can be taken only if it is continuous (no sharp steps) and if its 1st derivative is continuous.
► Wavefunctions must be continuous and have continuous 1st derivatives.
► Because of these restrictions, acceptable solutions to the Schrodinger wave equation do not result in arbitrary values for the energy of a system.
► The energy of a particle is quantized. It can have only certain energies.

15. Expectation value
► Let’s find the average score on a quiz. There were 5 problems on the quiz, worth 20pts each (no partial credit). The scores for 10 students are given below.
80, 80
On this particular quiz, 1/5 of the students received a score of 80. On future similar quizzes, we could say the probability of getting a score of 80 is 1/5.
60, 60, 60, 60
40, 40, 40 20
What is the average score? 54

16. Expectation value
80, 80
60, 60, 60, 60
40, 40, 40 20
► We calculated the average by multiplying each score by the probability of receiving that score and then found the sum of all those products.

17. Heisenberg’s uncertainty principle
► It is impossible to specify simultaneously, with arbitrary precision, both the momentum and the position of a particle.
► If the momentum of a particle is specified precisely, it is impossible to predict the location of the particle.
► If the position of a particle is specified exactly, then we can say nothing about its momentum.