# Undergraduate Tutorials

Having messed-up so comprehensively at a previous university which did operate a very personalised tutorial system, I felt second-time around that I deserved to do cold-turkey at a university that didn't.

But it should have done, and maybe does so nowadays. To get the best out of students you have to do more than simply lecture them; you have to help them overcome difficulties and misconceptions that can become a total block to further progress in a subject.

A fellow-student from those days who has recently got in touch again, has remarked that he had got virtually nothing out of our first-year Maths lecture course, until in desperation before the end-of-year exams he had asked for a one-hour one-to-one consultation with one of the lecturers, the estimable Erwin Kronheimer, who explained all the tricky bits and sent him on his way rejoicing.

I had a complete mental block about fundamental aspects of what in those days was called "wave-mechanics". Were we to regard the electron, for chemical purposes at least (I wasn't seeking Absolute Truth, just a usable mental model), as a wave, a particle or (as the Banesh Hoffman book called it) a wavicle?

And, when we got to the Heisenberg Uncertainty Principle, my brain went into spasm. The text-books of those days devoted pages to explaining how Nature always managed to confound the experimentalist who tried to measure the exact dynamical attributes of an electron. This seemed to be all very ad hoc, and wouldn't it would be simpler to suppose that the scatter of results was due to intrinsic randomness rather than the intervention of quantum gremlins.

There was no way one could collar our lecturer, a very nice guy, but who was anyway an electrochemist rather than a theoretical wallah; so together with a fellow-student I wrote to Werner Heisenberg, the celebrated Nobel Laureate, and probably the most iconic physicist in the world at that time. And to our delighted surprise, he replied with a very courteous and helpful letter.

Here's the sort of pep-talk a decent tutor could have provided:

Very briefly, the rules of quantum mechanics as conventionally presented say that a measurement of some property p of a system forces the **state** of the system into an **eigenstate** of p, and the measured value obtained for p will be the **eigenvalue** corresponding to that eigenstate. This process is rather melodramatically described as the collapse of the **wavefunction** (a colloquial alternative to state-function).

But which **particular** eigenvalue **initially** prevails is **totally random**, although repeated measurements of p in the new state will reassuringly yield the **same** value – the system has **settled**, so to speak, into that particular eigenstate.

Subsequent measurement of a different property q won't upset the applecart as regards the value already obtained for p, **provided** that p and q are **mutually compatible**: technically speaking, providing they **commute**.

But if p and q **don't commute**, then measurement of q will produce an eigenstate of q that certainly won't be an eigenstate of p. Your existing value of p has just been b*ggered: more technically, you can't obtain simultaneously valid values of non-commuting properties.

**So far, so bad**. But it gets **much better** if we remember that only very rarely if at all do we perform measurements on a single system (electron, atom or molecule, for instance). We generally have an everyday-sized box, bottle or tank of them – a **macroscopic ensemble**. Our measurement procedure now gives us a weighted average of p values, a statistical average of all possible eigenvalues of p each weighted by the relative proportion of wavefunctions in the ensemble that have collapsed into that particular eigenstate. The average is called the **expectation value** of p.

And, remarkably, the rules of quantum mechanics also tell us exactly how this expectation value can be **calculated** quite independently of any measurement. In principle, if the relative proportions could be tabulated against the corresponding eigenvalues, the **statistical weighted mean** would equal the **expectation value** as independently calculated.

And if the same figures were to be graphed, for both p and q, their **statistical standard deviations**, known as the **uncertainties** in p and q respectively, would obey the famous **Heisenberg Uncertainty Principle**, in that the value of their product can never be less than ћ/2, where ћ is the reduced **Planck constant** h/2π.

The HUP is popularly presented in terms of p = momentum and q = position, but there are plenty of other pairs of non-commuting **conjugate properties** that are **mutually uncertain** in a similar way. Their dimensions (ie generalised units of measurement) are always such that their product has the dimensions of **action** (ie **energy x time**) in conformity with the units of h itself.