1 HEINSENBERG’S UNCERTAINTY PRINCIPLE “It is impossible to determine both position and momentum of a particle simultaneously and accurately. The product of uncertainty involved in the determination of position and momentum simultaneously is greater or equal to h/2Π ” Significance: “Probalility” replaces “Exactness” An event which is impossible to occur according to classical physics has a finite probability of occurrence according to Quantum Mechanics Heisenberg

The Uncertainty Principle Since we deal with probabilities we have to ask ourselves: “How precise is our knowledge?” Specifically, we want to know Coordinate and Momentum of a particle at time t = 0 – If we know the forces acting upon the particle than, according to classical physics, we know everything about a particle at any moment in the future

But it is impossible to give the precise position of a wave A wave is naturally spread out Consider the case of diffraction Most of the energy arriving at a distant screen falls within the first maximum The Uncertainty Principle

Can we know Coordinate and Momentum (velocity) at some time t = 0 exactly, if we deal here with probabilities? The answer in Quantum Mechanics is different from that in Classical Physics, and is presented by the Heisenberg’s Uncertainty Principle

Classical Uncertainty d Consider classical diffraction Most of light falls within first maximum The angular limit of the first maximum is at the first zero of intensity which occurs at an angle  set by the condition, d sin  =, so we can say that the angle of light is between +  and -  Consider the following: As the uncertainty in y increases the uncertainty in the y-component of the k-vector decreases

Classical Uncertainty The classical uncertainty relation

The Uncertainty Principle An experiment cannot simultaneously determine a component of the momentum of a particle (e.g., p x ) and the exact value of the corresponding coordinate, x. The best one can do is

The Uncertainty Principle 1.The limitations imposed by the uncertainty principle have nothing to do with quality of the experimental equipment 2.The uncertainty principle does imply that one cannot determine the position or the momentum with arbitrary accuracy – It refers to the impossibility of precise knowledge about both: e.g. if Δx = 0, then Δ p x is infinity, and vice versa 3.The uncertainty principle is confirmed by experiment, and is a direct consequence of the de Broglie’s hypothesis

HOWEVER Since the wavefunction, Ψ(x,t), describes a particle, its evolution in time under the action of the wave equation describes the future history of the particle – Ψ(x,t) is determined by Ψ(x, t = 0) Thus, instead of the coordinate and velocity at t = 0 we want to know the wavefunction at t = 0 – Thus uncertainty is built in from the beginning and the wavefunction at all times is related to the evolution of probability

Examples: Bullet p = mv = 0.1 kg × 1000 m/s = 100 kg·m/s – If Δp = 0.01% p = 0.01 kg·m/s – Which is much more smaller than size of the atoms the bullet made of! – So for practical purposes we can know the position of the bullet precisely

Examples: Electron (m = 9.11× kg) with energy 4.9eV Assume Δp = 0.01% p – Which is much larger than the size of the atom! – So on atomic scale uncertainty plays a key role

Quantum Mechanics The methods of Quantum Mechanics consist in finding the wavefunction associated with a particle or a system Once we know this wavefunction we know “everything” about the system!

The Uncertainty Principle Between energy and time

Physical Quantity  Operators: “formal form”  “actual operation” Momentum Total Energy Coordinate Potential Function Quantum Mechanical Operators

Wave Function of Free Particle Since the de Broglie expression is true for any particle, we must assume that any free particle can be described by a traveling wave, i.e. the wavefunction of a free particle is a traveling wave For classical waves:

Wave Function of Free Particle However, these functions are not eigenfunctions of the momentum operator, with them we do not find, But see what happens if we try, After some “manipulations” we get

Wave Function of Free Particle Recall that Thus, the wavefunction of a Free Particle This wave function is an eigenfunction of momentum and energy!

Expectation Values Only average values of physical quantities can be determined (can’t determine value of a quantity at a point) These average values are called Expectation Values – These are values of physical quantities that quantum mechanics predicts and which, from experimental point of view, are averages of multiple measurements Example, [expected] position of the particle

Expectation Values Since P(r,t)dV=|Ψ(r,t)| 2 dV, we have a way to calculate expectation values if the wavefunction for the system (or particle) is known In General for a Physical Quantity W – Below Ŵ is an operator (discussed later) acting on wavefunction Ψ(r,t)

Expectation Value for Momentum of a Free Particle Generally Free Particle

Properties of the Wavefunction and its First Derivative 1.must be finite for all x 2.must be single-valued for all x 3.must be continuous for all x