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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. sin x or e x 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 2 d 2 2m dx 2 2 2 2m where V( x ) E for a 1-D system V E for a 3-D system 2 2 2 2 2 2 x y z for Cartesian coordinate 2 2 1 2 2 2 r rr r for spherical coordinate 2 2 System with Spherical Symmetry 2 2 2 1 2 2 V E 2 rr r 2m r where 1 2 1 sin sin 2 2 sin 2 2 2 2 1 1 2 1 V E r 2 rr r 2 sin 2 2 sin sin 2m Eigenstate ► in general, the Schrodinger eqn is H E ► H is the Hamiltonian operator, i.e., the energy operator. E H ► Schrodinger eqn is an eigenvalue equation (operator)(function) = (constant) x (same function) eigenvalue eigenfunction Operator ►(operator) (function) = (constant) x (same function) ̂f f The factor is called the eigenvalue of the operator ̂ ► The function f (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. pˆ i d dx ► If we want to find the linear momentum of a particle in the x direction, we use the following eigenvalue equation. pˆ p ► The eigenvalue, p, is the momentum. Correspondence ► The kinetic energy operator is then created from the momentum operator. 2 p2 mv 2 mv Ek 2m 2 2m 2 2 2 ˆ p d Eˆ k H 2m 2m dx 2 ► If we want to find the kinetic energy of a particle in the x direction, we use the following eigenvalue equation. H E 2 d 2 E 2m dx 2 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. 2 or * Interpretation 2 2 2 or * probability density or * no negative or complex probability densities or * written 2 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%. N 2 2 dx 1 1 N 2 dx 1 2 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 ˆ ˆ d * ► 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 60, 60, 60, 60 40, 40, 40 20 2 (80) 16 On this particular quiz, 1/5 of the students 4 (60) 24 received a score of 80. On future similar 10 quizzes, we could say the probability of 3 (40) 12 getting a score of 80 is 1/5. 10 10 1 (20) 2 10 What is the average score? 54 Expectation value 2 80, 80 10 (80) 16 (60) 24 10 3 (40) 12 10 1 (20) 2 10 4 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. ˆ ˆ d * 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. p x 2