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.

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.

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

System with Spherical Symmetry where

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

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)

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.

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

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.

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.

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.

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

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.

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.

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

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.

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.