Heisenberg Uncertainty Principle

• Heisenberg (1926) thought about measuring simultaneously the position and momentum (velocity) of an electron. Realization → measurement of both with precision is impossible and, in fact, the measurement process perturbs the system (electron, etc).

Heisenberg, Light and Electrons• Measuring the position of a very small particle

such as an e- requires the use of short wavelength light (high frequency, high momentum light (Compton effect)). High frequency light will alter the electron’s momentum. Use low frequency (long λ) light and the position of the electron is not well determined.

Heisenberg Principle – Equations:

• The Heisenberg uncertainty principle has several forms. We denote the uncertainty in momentum, for example, by Δp. In one dimension we have

• ΔpxΔx ≈ h

• If q denotes a three dimensional position coordinate then

• ΔpΔq ≥ h/4π

Heisenberg and Mass

• Since momentum p = mu (u = velocity) the Heisenberg expression can be written as

• ΔuΔq ≥ h/4πm• This equation tells us that the product of the

uncertainties in velocity and position decrease as the mass of the particle increases. Ignoring history, Heisenberg tell us that as the mass of a particle/system increases we move towards a deterministic description (Newton!).

Heisenberg and Spectroscopy• The Heisenberg expression can be recast, in

terms of energy and time, as• ΔEΔt ≥ h/4π• Relates in “real life” to widths of spectral lines.

In electronic emission spectra Δt can be small (short lifetime in the excited state). This means that ΔE will be “large”. The transition energy covers a range of values – the observed spectral “line” is broadened.

de Broglie, Electrons and Waves

• Einstein’s theory of relativity tells us that the energy of a particle having a momentum p and rest mass m is given by

• E2 = p2c2 + m2c4

• If our interest lies in photons (zero rest mass) then

• Ephoton = pc = pλν

De Broglie – continued:

• Using the familiar expression from Planck, E = hν, we get for photons ( i.e., light or electromagnetic radiation).

• λ = h/p• De Broglie suggested that this last expression

could apply to beams of particles with a finite rest mass, m, and a velocity u (momentum p = mu).

De Broglie and Diffraction:

• For finite rest mass particles de Broglie gives us

• λ = h/p = h/mu• This expression does have meaning in our

physical world. Wave properties of subatomic particles and light molecules have been demonstrated though , for example, a variety of diffraction experiments.

de Broglie – examples:

• 1. Electrons are easily accelerated by an electric field. Find the de Broglie wavelength for an electron accelerated by a voltage of 12.5 kilovolts.

• 2. Would the de Broglie picture be more useful in describing the behaviour of a neutron or an NFL lineman?

De Broglie and Molecular Structure:

• Aside: Highly accelerated and collimated electrons are diffracted when they move through a gas. Knowing the energy or de Broglie wavelength of the collimated electrons one can, for simple gas phase molecules, use the diffraction pattern obtained to determine a three dimensional structure for the molecules. (Why are protons and neutrons less useful for such experiments?).

de Broglie, Waves and Electrons:

• The work of de Broglie and others showed that light and subatomic particles had something in common – wave properties. This led Schrodinger, in particular, to wonder whether equations used to describe light waves could be modified to describe the behaviour of electrons. This led to quantum mechanics and a probabilistic description of the behaviour of electrons.

Operators:

• A brief class discussion of operators, eigenvalues, eigenfunctions and eigenvalue equations is needed before moving to, for example, the postulates of quantum mechanics. Some of this material may be familiar from mathematics courses. The “eigenfunctions” that are useful in describing particles with wave properties are of familiar form (and, in part, predictable?).

Postulates of Quantum Mechanics:

• The development of quantum mechanics depended on equations that are not, in the normal sense, derivable. This development was based on a small number of postulates. The reasonableness of these postulates will become clear through application of the postulates.

Quantum Mechanics – Postulates:

• Postulate 1: A quantum mechanical system or particle can be completely described using a wave function, ψ. (Wave functions were introduced briefly in Chemistry 1050).

• In different examples the problems of interest will have varying dimensionality. Correspondingly, one sees wave functions described in terms of one or more coordinates/variables.

Postulate 1 – Dimensionality:

• Possible “forms” of wave functions:• One dimensional problems: ψ(x), ψ(r), ψ(θ)• Two dimensional problems: ψ(x,y) etc.• Three dimensional problems: ψ(x,y,z), ψ(r,θ,φ)• If the time dependant evolution of the system

needs to be treated (or, appears interesting!) the wave functions will have the form ψ(x,y,z, t) or ψ(r,θ,φ,t).

Postulate 1 – Wave function Properties:

• The wave function ψ(x,y,z) (for example) must be continuous, single valued (not a new requirement!) and square integrable. Both real and imaginary wave functions are encountered. Thus, if ψ(x) is imaginary (for example, eimx , where m is an integer) then we will need the complex conjugate wave function ψ*(x) = e-imx.