Relationship between a material wave and the uncertainty of position and momentum

[John Constantine]

TL;DR Summary

Relationship between the wavelength of a material wave and the uncertainty of position and momentum.

I'm just an ordinary person who's very interested in physics. I'm posting a question because I'm curious about quantum mechanics.

The wavelength of the material wave that can be obtained when a baseball with a mass of 150 g is thrown at 40 m/s is 1.1×10^-34m by the h/mv formula. As you can see from the numbers, the wavelength is too small for the atomic unit. It cannot be detected by humans or most measuring instruments.

As far as I know, the wavelength of matter in a macroscopic object(ex. table, baseball, etc..) is very short. The shorter the wavelength of a material wave, objects have more particle properties than wave properties. It's understandable that a macroscopic object has particle properties and that the wavelength of the object's material waves is very short. Most of the objects we face in our daily lives can be fairly accurate at the same time as their position and momentum.

However, macroscopic objects do not defy the principle of uncertainty. Obviously, the more accurate you try to locate the object, the more inaccurate the momentum will be.

I'll ask you a question now.

1. For macroscopic objects, the wavelength of the material wave is extremely short. If the wavelength of a material wave is extremely short, does the inaccuracy of position and momentum decrease at the same time than if the wavelength is long?

2. If Question 1 is wrong, is a macroscopic object measured more accurately than a microscopic object, either position or momentum?

3. What does it mean that macroscopic objects have less inaccuracy in position and momentum compared to microscopic objects?

 

[anuttarasammyak]

Heisenberg Uncertainty relation
$$\triangle x \triangle p \ge \hbar/2$$
and de Broglie relation
$$\lambda=\frac{h}{p}$$
give us the relation
$$\triangle x \ge \frac{\lambda^2}{4\pi \triangle \lambda}$$
You may estimate RHS for macro objects. Not only value of $\lambda$ but also its variance $\triangle \lambda$ matters.

 

[John Constantine]

give us the relation
$$\triangle x \ge \frac{\lambda^2}{4\pi \triangle \lambda}$$
You may estimate RHS for macro objects. Not only value of $\lambda$ but also its variance $\triangle \lambda$ matters.

I'm sorry to bother you. I can only do high school level math. Can you briefly explain what that relation means?

 

[PeroK]

I'm sorry to bother you. I can only do high school level math. Can you briefly explain what that relation means?

Note that the De Broglie matter wave theory is not part of QM. It was an early version of QM that was superseded by modern QM in the 1920s.

 

[PeroK]

3. What does it mean that macroscopic objects have less inaccuracy in position and momentum compared to microscopic objects?

If you apply the uncertainty principle to a baseball, then the uncertainty in position and/or momentum is immeasurably small. What does it mean for the uncertainty in its position to be a millionth of the diameter of an atom? That's not something even a major league batter is going to notice!

 

[gentzen]

Note that the De Broglie matter wave theory is not part of QM. It was an early version of QM that was superseded by modern QM in the 1920s.

Why do you write this? I would say that De Broglie matter wave theory is consistent with modern non-relativistic QM, and can be derived from it (via invariance under canonical transformations), and thereby interpreted suitably. And it wasn't "early" either (quoted from wikipedia):

It remained to extend the wave considerations to any massive particles, and in the summer of 1923 a decisive breakthrough occurred. De Broglie outlined his ideas in a short note "Waves and quanta" (French: Ondes et quanta, presented at a meeting of the Paris Academy of Sciences on September 10, 1923), which marked the beginning of the creation of wave mechanics.

 

[PeroK]

Why do you write this?

I learned QM from the textbooks by Griffiths and Sakurai. There was no mention of De Broglie matter waves in either. Except one historical footnote in Griffiths.

 

[vanhees71]

The problem with de Broglie's matter waves is that he didn't provide a wave equation for them. Famously this was triggered by a somewhat ironic statement by Peter Debye in a colloquium delivered by Schrödinger at the University of Zürich, where he stated that Schrödinger should provide a wave equation when talking about de Broglie waves, and that's what Schrödinger famously did only some months later in 1926.

 

[anuttarasammyak]

I'm sorry to bother you. I can only do high school level math. Can you briefly explain what that relation means?

The relation is written as
$$\triangle x \ge (4\pi)^{-1}\frac{\lambda}{\frac{\triangle \lambda}{\lambda}}$$
where
$\lambda$ is average wavelength of de Broglie waves which consists "wave packet" of particle
$\frac{\triangle \lambda}{\lambda}$ is standard deviation / average of the wavelength.

Say keeping $\frac{\triangle \lambda}{\lambda}$ constant, the minimum $\triangle x$ is proportional to $\lambda$ , average wavelength of de Broglie waves, which has small value for a macro body in motion even when it moves in extremely low speed.

 

[gentzen]

I learned QM from the textbooks by Griffiths and Sakurai. There was no mention of De Broglie matter waves in either. Except one historical footnote in Griffiths.

At some (rather late) point I sat down and read "Tutorium Quantenmechanik" by Jan-Markus Schwindt cover-to-cover. (Which means I didn't manage to read any other introduction to QM cover-to-cover before.) On page 125 in "Streber-Ecke 3.2" he explains (actually "Nerd’s Corner 3.2" on page 114 in the English version):

A canonical transformation is a transformation $$x_i→x_i'({\bf r},{\bf p},t), \quad p_i→p_i'({\bf r},{\bf p},t), \qquad (3.227)$$ for which the Poisson brackets (3.225) still hold in the new variables. This property is inherited to the commutators of the associated operators. One can then rewrite the wave function as a function of the new variables $\{x_i'\}$. The operators $\{P_i'\}$ then act again via $\partial/\partial x_i'$.

Of course, he cheats with those "Nerd’s Corners," because this allows him to introduce concepts and claims which would be very hard to introduce or prove rigorously. For examples, a canonical transformation can be non-linear in general, but it is unclear how such a non-linear transformation would act in the quantum case, and whether his claim would still be correct. Of course, he just uses linear canonical transformations in his examples

Exercise 3.16
Show that in one-dimensional space the transformation $$x \to x'=p, \quad p \to p'=-x \qquad (3.228)$$ is canonical. In this way, find a deeper reason for the behavior of and in momentum space, (3.185).

and Griffiths in "5.1 Zwei-Teilchen-Systeme" in "Aufgabe 5.1" also just uses a linear canonical transformation, when he transforms to center-of-mass + relative coordinates. And this linear canonical transformation into center-of-mass + XXX is the derivation of the de Broglie relation from modern non-relativistic QM which I have in mind. The "corresponding suitable interpretation" I have in mind then restricts itself to statements about the center-of-mass, and the total momentum.

 

[vanhees71]

Well, there are some good but many bad textbooks...

 

[hutchphd]

The problem with de Broglie's matter waves is that he didn't provide a wave equation for them.

There was no mention of De Broglie matter waves in either.

With respect, these both seem irrelevant to the specific question at hand. The relationship follows from Fourier (wave) synthesis and requires only the wave nature and the DeBroglie Hypothesis.
Of course De Broglie is incomplete.