UNIT 1 Quantum Mechanics

Schrödinger equation 1926, Erwin Schrödinger (Austria) Describe a particle with wave function Wave function has full information about the particle

The Born interpretation of the Wave Function Contains all the dynamic information about the system Born made analogy with the wave theory of light (square of the amplitude is interpreted as intensity – finding probability of photons) Probability to find a particle is proportional to It is OK to have negative values for wave function Max Born Probability Density

The Born interpretation of the Wave Function

The Born interpretation of the Wave Function

where i is the square root of -1. The Schrödinger wave equation in its time-dependent form for a particle of energy E moving in a potential V in one dimension is: where i is the square root of -1. The Schrodinger Equation is fundamental equation of Quantum Mechanics. where V = V(x,t)

General Solution of the Schrödinger Wave Equation when V = 0 The solution This works as long as: which says that the total energy is the kinetic energy.

General Solution of the Schrödinger Wave Equation when V = 0 In free space (with V = 0), the general form of the wave function is which also describes a wave moving in the x direction. In general the amplitude may also be complex. The wave function is also not restricted to being real. Only the physically measurable quantities must be real. These include the probability, momentum and energy.

Normalization and Probability The probability P(x) dx of a particle being between x and x + dx is given in the equation The probability of the particle being between x1 and x2 is given by The wave function must also be normalized so that the probability of the particle being somewhere on the x axis is 1.

Properties of Valid Wave Functions Conditions on the wave function: To avoid infinite probabilities, the wave function must be finite everywhere. The wave function must be single valued. The wave function must be differentiable. This means that it and its derivative must be continuous. (An exception to this rule occurs when V is infinite.) In order to normalize a wave function, it must approach zero as x approaches infinity.

Time-Independent Schrödinger Wave Equation The potential in many cases will not depend explicitly on time. The dependence on time and position can then be separated in the Schrödinger wave equation. Let: which yields: Now divide by the wave function y(x) f(t): The left side depends only on t, and the right side depends only on x. So each side must be equal to a constant. The time dependent side is:

Time-Independent Schrödinger Wave Equation This equation is known as the time-independent Schrödinger wave equation, and it is as fundamental an equation in quantum mechanics as the time-dependent Schrodinger equation. So often physicists write simply: where: is an operator.

Eigenvalues and eigenfucntions Eigenvalue equation (Operator)(function) = (constant factor)*(same function) Operator Eigenfunction Eigenvalue Solution : Wave function Allowed energy (quantization)

Expectation Values The expectation value, , is the weighted average of a given quantity. In general, the expected value of x is: If there are an infinite number of possibilities, and x is continuous: Quantum-mechanically: And the expectation of some function of x, g(x):

Momentum Operator To find the expectation value of p, With k = p / ħ we have This yields This suggests we define the momentum operator as The expectation value of the momentum is

Position and Energy Operators The position x is its own operator. Energy operator: The time derivative of the free-particle wave function is Substituting w = E / ħ yields The energy operator is: The expectation value of the energy is:

Operators Position Momentum Potential energy Kinetic energy Total energy

Quantum Numbers Definition: specify the properties of atomic orbitals and the properties of electrons in orbitals There are four quantum numbers The first three are results from SchrÖdinger’s Wave Equation

Orbital Quantum numbers An atomic orbital is defined by 3 quantum numbers: n l ml Electrons are arranged in shells and subshells of ORBITALS . n  shell l  subshell ml  designates an orbital within a subshell

Quantum Numbers Symbol Values Description n (major) 1, 2, 3, .. Orbital size and energy = -R(1/n2) l (angular) 0, 1, 2, .. n-1 Orbital shape or type (subshell) ml (magnetic) -l..0..+l Orbital orientation in space Total # of orbitals in lth subshell = 2 l + 1

Quantum Theory Model Orbitals One “s” orbital Three “p” orbitals Five “d” orbital

Ehrenfest’s theorem Ehrenfest’s theorem states that quantum mechanics provides the same results as classical mechanics for a particle for which the average or expectations values of dynamical quantities are involved.