Presentation on theme: "Schrodinger wave equation"— Presentation transcript:
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 systemfor a 3-D systemwherefor Cartesian coordinatefor spherical coordinate
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)eigenvalueeigenfunction
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 densitieswrittenBoth 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, 80On 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, 6040, 40, 4020What is the average score?54
16 Expectation value80, 8060, 60, 60, 6040, 40, 4020► 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.