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Source: D. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 2004) R. Scherrer, Quantum Mechanics An Accessible Introduction (Pearson Int’l Ed., 2006) R. Eisberg & R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (Wiley, 1974). LECTURE 1. Topics Today.
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Source: D. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 2004) R. Scherrer, Quantum Mechanics An Accessible Introduction (Pearson Int’l Ed., 2006) R. Eisberg & R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (Wiley, 1974) LECTURE 1
Topics Today • Quantum Mechanics and Classical Physics. • Waves: Plane Waves, Wave Packets. • Wavefunction – Properties, Normalization, Expectation Values, Scroedinger Equation. • Operators • Eigenvalues and Eigenfunctions. • Uncertainty Principle.
Classical Physics • Classical Mechanics • Electricity and Magnetism • Thermodynamics Modern Physics: Areas of Physics emerging from Quantum Mechanics • Atomic Physics • Nuclear Physics • Particle Physics • Condensed Matter Physics
THE WAVE FUNCTION Wavefunction of a physical system contains the measurable information about the system. Quantum Mechanics Approach Particle Wave Function: Y(x,t) Y (x,t) is obtained by solving Schroedinger Equation: V(x) = Potential Energy
Probability in Quantum Mechanics WavefunctionY: The probability amplitude for finding a particle at a given point in space at a given time. Actual probability : YY* The sum of the probabilities for all of space must be equal to Normalization of Wave Functions: To obtain the physically applicable probability amplitudes.
Operators • To obtain specific values for physical parameters, the quantum mechanical operator associated with that parameter is operated on the wavefunction. When an operator operates on the wave function, a number is obtained. This number corresponds to a possible result of a physical measurement of that quantity. Operators in Quantum Mechanics are linear operators. If P is a linear operator, where a and b are constants.
Operators in Quantum Mechanics Observables Operators
Expectation Value For a physical system described by awavefunctionY , the expectation value of any physical observable q can be expressed in terms of the corresponding operator Q as follows: The wavefunction must be normalized and that the integration is over all of space. The function can be represented as a linear combination of eigenfunctions of Q, and the results of the operation gives the physical values times a probability coefficient. Expectation value gives a weighted average of the possible observable values. (Standard Deviation)2=
Expectation Values Expectation Value: To relate a quantum mechanical calculation to something you can observe in the laboratory. For the position x, the expectation value is defined as Expectation Value of x: The average value of position for a large number of particles which are described by the same wavefunction. Example: The expectation value of the radius of the electron in the ground state of the hydrogen atom is the average value you expect to obtain from making the measurement for a large number of hydrogen atoms.
Expectation Value of Momentum The expectation value of momentum involves the representation of momentum as a quantum mechanical operator. Where is the operator for the x component of momentum.
Hamiltonian The Hamiltonian contains the operations associated with the kinetic and potential energies and for a particle in one dimension can be written: The Hamiltonian operator also generates the time evolution of the wavefunction in the form The full role of the Hamiltonian is shown in the time dependent Shrodinger equation where both its spatial and time operations manifest themselves. Y • The operator associated with energy is the Hamiltonian, and the operation on the wavefunction is the Schrodinger equation.
Eigenvalues and Eigenfunctions • Solutions exist for the time independent Schrodinger equation only for certain values of energy, and these values are called "eigenvalues*" of energy. • Corresponding to each eigenvalue is an "eigenfunction*". • The solution to the Schrodinger equation for a given energy involves also finding the specific function which describes that energy state. • The solution of the time independent Schrodinger equation takes the form
The eigenvalue concept is not limited to energy. When applied to a general operator Q, it can take the form if the function is an eigenfunction for that operator. The eigenvalues qi may be discrete, and in such cases we can say that the physical variable is "quantized" and that the index i plays the role of a "quantum number" which characterizes that state.
The Wavefunction • Each particle is represented by a wavefunction, Y(position, time). • Y*Y = The probability of finding a particle at that time. • Wavefunction is the solution of the Schroedinger Equation. • Schroedinger Equation plays the role of Newton’s Law in classical mechanics. It predicts the future behaviour of a dynamic system. It predicts analytically and precisely the probability of events or outcome. The detailed outcome depends on chance. Large number of events: The Schroedinger Equation predicts the distribution of results.
The Uncertainty Principle • The position and momentum of a particle cannot be simultaneously measured with arbitrarily high precision. • There is a minimum for the product of the uncertainties of these two measurements.
