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 havecertainly heard of this important relationship.It states:

Here, x is the “uncertainty of the particle position”– in other words, the precision with which the particleposition can be determined (the uncertainty we 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 mo-mentum” – or, the precision with which the momentumcomponent in the x direction can be measured.

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(where hpx x

The Heisenberg Uncertainty Relationship states that the position x and the momentumpx 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 positiondetermination means that the momentum can-not be precisely determined – and vice versa.

The two uncertainties are such that theirproduct will always be . And this is not because our apparatus is not perfect. Thisis a LAW OF NATURE.

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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 weregiving us that information “like a rabbit from amagician’s tall hat”– it is so, you have to believe!

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Finally, at last term of my junior year, I wastaking a “Quantum Mechanics One” course,and only then the professor showed us howto derive the Heisenberg Uncertainty Relationfrom 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 whenI had to believe in the HUR only “because wisemen had shown that it is so”.

Therefore, my sincere wish is that my studentsnever have such odd feelings – and therefore Ialways 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 hasto 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 Mechanicsshow this method, and they certainly think it is “general enough”because they don’t discuss the method based on the Schwartzinequality.

With wavepackets, there is a “trade-off”: to get a narrowone, 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 explanationsonly. The material is given here for youto know that the final result CAN BE deri-ved 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 al-ways may check, if you change your mind.

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If we think of the standard deviations as of“uncertainties”, and we use for them thesymbols k and x , we can write:

But for the de Broglie waves the wavenumberand the momentum are related as:

Which leads to the final result:

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The “Gaussian wave packet” is also knownas “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.

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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 movesperiodically 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 kindassociated with the de Broglie waves.

THEN, WHAT?!

Well, to answer this question, we have to clear upcertain 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 par-ticles we observe. We see tiny specs on a photographicfilm, tiny flashes on a fluorescent screen, “tracks” in acloud or bubble chamber.

Wave-like properties of particles are always manifestedindirectly.

Now, consider the double slit experiment with electrons.Extremely valuable for our understanding of de Brogliewaves are experiments in which electrons are shone atthe double-slit apparatus one at a time, which enablesus to see individual flashes on a fluorescent screen.

In a famous experim-ent done in Bologna,Italy, in 1974, the fla-shes from electrons reaching the screen were detected by anultrasensitive photo-detector, and aftereach flash the film In a camera focused on the screen wasadvanced by one frame. Then, the datawere combined, showing the time evolutionof the interference pattern.

In an analogous experiment done in Japan in1989 a more modern technique of electronicrecording 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), phy-sicists finally found an answer that was accep-ted by a broad majority: namely, that the de Bro-glie waves are “waves of probability” – meaningthat the value of the de Broigle wavefunction ofa particle expresses the probability of findingthe particle at a given point:

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finding ofy Probabilit 2)(xIt should be noted that Albert Einstein did not like the “probabilisticinterpretation”, and he died unconvinced. Although the majority ofphysicists now accept this interpretation, the discussion is not yetfinished – the problem is still the subject of ontological debate.