Week VIII Quantum Mechanics Philosophy of Physics Week VIII Quantum Mechanics

Why Quanta? The Old Quantum Theory Classical physics: 17th – 19th century Various problems, but thought to be small (aether, radiation, radium, etc.) Old quantum theory: 1900 – 1925 Hodge-podge of results Planck, Einstein, Bohr

Max Planck (1858-1947) Founder of quantum theory 1900 Introduced “Planck’s constant” h to make sense of “black-body radiation” h = 6.626068 × 10-34 m2 kg / s ℏ = h/2π Early champion of Einstein Father figure in German science

Albert Einstein (1879-1955) In 1905 wrote three great papers Special relativity Brownian motion Light quanta Light quanta (now called photons) were introduced to explain the photo-electric effect (Nobel Prize for this, not relativity) Made use of Planck’s constant and introduced the idea that light had both particle and wave aspects.

Niels Bohr (1885-1962) Danish, ranks with Einstein in importance in 20th century Created “Bohr atom” in 1913, using Planck’s quantum of action. Electron orbits come in steps (ie, quantized). Explained nature of atoms and spectral properties.

Quantum Mechanics (QM) The new quantum theory arose 1925-27 (unchanged since then) Version 1: Matrix mechanics created by Heisenberg, Born, Jordan, etc. Version 2: Wave mechanics created by de Broglie, Schrödinger, etc. (Pronounced: de-broy) Two versions shown to be equivalent Others include: Pauli, Dirac, Bohr, von Neumann, etc.

Werner Heisenberg (1901-1976) Created “matrix mechanics” in 1925 “Uncertainty Principle” Strong positivist / empiricist inclinations in early career Later came to hold an “Aristotelian” view of QM

Erwin Schrödinger (1887-1961) Created “wave mechanics” in 1926 Showed (surprisingly) that it is equivalent to matrix mechanics. Like Einstein, he was a realist and was opposed the interpretation of QM offered by Bohr, etc. “Schrödinger’s cat” aims to show problems of Bohr’s view.

Quantum Mysteries When randomly polarized light hits a polaroid filter, half passes through. A second filter at right angles stops the remaining light. Hypothesis: Light came in two types: the vertically polarized light was stopped by the vertical filter and the horizontally polarized light was stopped by the horizontal filter. (I = intensity of light). Could this hypothesis be right?

So far, so good. But look what happens when we put an intermediate filter between the vertical and the horizontal filters. Some light makes it through. How do we explain that?

More mysteries Suppose we fired bullets through one of two holes to a back screen. P1 = the probability of going through hole 1 and landing at a particular point at the back screen. P2 = same for hole 2. P12 = P1 + P2 = probability of going through either hole 1 or hole 2 and hitting the point on the back screen. (Diagram and next two from Feynman, Lectures on Physics, vol III, ch 1)

So far, what happens is just common sense. Here’s the same idea with water waves. Different pattern than bullets, but it’s just what we expect with waves.

Now, the same again with electrons. Electrons are like bullets, ie, point particles, so we should expect something like the first outcome (bullets). But we get something like the second (waves). What’s going on?

What is an interpretation of QM? QM can explain the two mysterious examples that I just described in the sense that the observed outcomes are exactly as QM predicts they would be. In this regard, QM is a fabulously successful theory. But still we are puzzled. We want not only accurate predictions, we also want some sort of understanding. We shall now look at the theory in a bit of detail, then try to make sense of it, that is, try to interpret it.

But what is an interpretation? A good interpretation would tell us how the world is and why the formalism of QM works so well. Is this possible? Feynman says NO; just learn how to use QM, but don’t try to understand it – that’s impossible; no one can understand it. (“Shut up and calculate”) We’ll proceed on the assumption that Feynman is wrong – we can do more than calculate; we might be able to understand.

Main Principles of QM The mathematical formalism of QM distinguishes states (eg, state of an electron), properties (eg, momentum of the electron), and magnitudes (eg, the particular value of the momentum). Quantum properties are often called “observables” (even though they might not be observable at all). I will usually call them properties.

States We often use some sort of non-physical space to represent the state of a system. We say that at time t the unemployment rate is u. That tells us something about the state of the economy.

Quantum States A quantum system (an electron, a photon, etc.) is represented by a vector ψ in a Hilbert space H. A Hilbert space is a finite or infinite-dimensional vector space over the field of complex numbers. Any Hilbert space satisfies the following axioms. (The formalities, after the following diagram, are for math/physics students who have done some QM, but have not seen it presented axiomatically. Skip the strange formalities and focus on the discussions and the pictures, if you wish.)

Notation This is a sample; there are many variations. Regular Dirac State of system ψ, φ,… |ψ>, |φ>, … Eigenstate or eigenvector α, β, … α1, α2, … |α>, |β>, … |a1>, |a2>, … eigenvalues a, b, c, … a1, a2, … Inner product (ψ, φ) <ψ|φ>

Hilbert Space H is a Vector Space and C is the set of complex numbers

Principle: The representation of states

Linear operators

Principle: The representation of properties

Principle: Measurements are eigenvalues Principle: The Born rule

Principle: Heisenberg’s Uncertainty

Principle: The Projection Postulate

Principle: The Schrödinger equation

Composite systems

Light through a Polaroid filter