The Heisenberg Uncertainty Relationship (HUR)

The Heisenberg Uncertainty Relationship (HUR) (under construction. When expected to be ready? – it’s uncertain)

The Heisenberg Uncertainty Principle But one thing is not uncertain to me – you have certainly heard of this important relationship. It states: Here, x is the “uncertainty of the particle position” – in other words, the precision with which the particle position can be determined (the uncertaintywe are talking about now is NOT that resulting from the im- perfect measuring equipment; assume that it is “infi- nitely precise”). And px is the “uncertainity of the particle momentum” – or, the precision with which the momentum component in the x direction can be measured.

The Heisenberg Uncertainty Relationship states that the position x and the momentum px cannot be both determined with an urest- rictibly good precision. Even the best possi- ble apparatus would not help here: there is always a tradeoff! High precision in position determination means that the momentum cannot be precisely determined – and vice versa. The two uncertainties are such that their product will always be . And this is not because our apparatus is not perfect. This is a LAW OF NATURE.

A story from Dr. Tom’s own life experience: When I was an undergraduate student, we were told many times by our instructors on various occasions: “As shown by Heisenberg, ….” or: “As is well known, ….” or: “As the Uncertainty Relation states, ...” I was feeling frustrated, because they always were giving us that information “like a rabbit from a magician’s tall hat”– it is so, you have to believe!

Finally, at last term of my junior year, I was taking a “Quantum Mechanics One” course, and only then the professor showed us how to derive the Heisenberg Uncertainty Relation from the “first principles”. The procedure is based on a fundamental mathematical theo- rem called “the Schwartz Inequality”. But I still remember how frustrated I was when I had to believe in the HUR only “because wise men had shown that it is so”.

Therefore, my sincere wish is that mystudents never have such odd feelings – and therefore I always want to show them ASAP where this famous formula comes from. The method based on the “Schwartz Inequality” is too advanced for this course because one has to first get enough knowledge of the foundations of Quantum Mechanics. But there is a very instructive method of deri- ving the HUR for wavepackets composed of de Broglie Waves, and I want to show you that. Note: some authors of textbooks on introductory Quantum Mechanics show this method, and they certainly think it is “general enough” because they don’t discuss the method based on the Schwartz inequality.

With wavepackets, there is a “trade-off”: to get a narrow one, you have to take waves from a broad range of k ; and narrower range produces a wider packet:

Now there will be several pages of calcu- lations (mostly, integrals). We will not discuss this math step- by-step in class, but we will scroll through slides #10 to #18 with brief explanations only. The material is given here for you to know that the final result CAN BE derived in a fully rigorous manner – and if you wish, you may check! But it is not neces- sary, if you prefer to accept the results without proof, this is also OK. But you always may check, if you change your mind.

If we think of the standard deviations as of “uncertainties”, and we use for them the symbols k and x , we can write: But for the de Broglie waves the wavenumber and the momentum are related as: Which leads to the final result:

The “Gaussian wave packet” is also known as “the minimum uncertainty wave packet”. For a wave packet whose k spectrum is de- scribed by any other function than a Gaus- sian, the uncertainties are always such that: (please accept without proof). What we did here should not be treated as a “derivation of the Heisenberg relationship”– HUR can be derived from more fundamental assumptionts – it was only an illustration of “how the HUR works” in wave packets.

Last question – what is the physical Interpretation of de Broglie waves? In sound waves, it is the air molecules that oscillate; A sound wave consists of areas of higher and lower density (or pressure). In EM radiation, it is the electric and magnetic field that oscillate. In waves on water surface, it’s the water that moves periodically up and down. In seismic waves reaching the Earth’s surface – every- body knows, let’s better not talk about sad things.

But what is oscillating in de Broglie waves? What is the “undulating agent” in such waves? Certainly, nothing material! (the often used term “waves of matter” is highly misleading!) A field? – no, surely, there is no field of any kind associated with the de Broglie waves. THEN, WHAT?! Well, to answer this question, we have to clear up certain things.

Note that we always observe particles “as particles”. In an act of observation, or in an “act of particle detec- tion”, we never see a wave. Things we see are always “manifestations” of the particle-like nature of the particles we observe. We see tiny specs on a photographic film, tiny flashes on a fluorescent screen, “tracks” in a cloud or bubble chamber. Wave-like properties of particles are always manifested indirectly. Now, consider the double slit experiment with electrons. Extremely valuable for our understanding of de Broglie waves are experiments in which electrons are shone at the double-slit apparatus one at a time, which enables us to see individual flashes on a fluorescent screen.

In a famous experim- ent done in Bologna, Italy, in 1974, the flashes from electrons reaching the screen were detected by an ultrasensitive photo- detector, and after each flash the film In a camera focused on the screen was advanced by one frame. Then, the data were combined, showing the time evolution of the interference pattern.

In an analogous experiment done in Japan in 1989 a more modern technique of electronic recording was used – but the results were essentially the same as in Bologna.

To make the long story short: after many years debating and many disputes (often heated), physicists finally found an answer that was accep- ted by a broad majority: namely, that the de Bro- glie waves are “waves of probability” – meaning that the value of the de Broigle wavefunction of a particle expresses the probability of finding the particle at a given point: It should be noted that Albert Einstein did not like the “probabilistic interpretation”, and he died unconvinced. Although the majority of physicists now accept this interpretation, the discussion is not yet finished – the problem is still the subject of ontological debate.

The Heisenberg Uncertainty Relationship (HUR)