How do Heisenberg Uncertainty & de Broglie wavelength apply to atom trap?

[natalie.*]

I had this https://www.physicsforums.com/showthread.php?p=3200140#post3200140", which I posted on PF. I got the answer, but then I started thinking more about it and have some theoretical questions.

If you did have this particle of mass m in a box of length L, which you are trying to stop, you know that by the Heisenberg Uncertainty Principle,$$\Delta x \Delta p_{x}\geq\frac{h}{2}$$, you can only know it's velocity within the range $$\Delta v=\frac{h}{2Lm}$$. That means (the answer to the homework question) the velocity of the particle must be in the range $$-\frac{1}{2}*\frac{h}{2Lm}\leq v \leq \frac{1}{2}*\frac{h}{2Lm}$$ (I think coming to this conclusion is where there's a problem but I'm not sure).

According to the whole de Broglie wave-like nature of the particle thing though, it only has discrete, quantized, allowable energies, and therefore velocities, according to $$v_{n}=n\frac{h}{2Lm}$$, which looks a lot like the equation from the Heisenberg Uncertainty Principle, but that equation, where n is a natural number, says that the lowest possible speed of the particle must be $$\frac{h}{2Lm}$$.

I get that for the sake of the question you're just ignoring the matter wave thing, but I just want to reconcile this in my head. The two formulas obviously look so similar, but I just can't make them work together. How should I be thinking about it?

 

[AndyW]

Thank you for bringing up these theoretical questions. It is important to think critically about the concepts we learn in physics, and I am happy to help you reconcile the two formulas you mentioned.

Firstly, it is important to note that the Heisenberg Uncertainty Principle and the de Broglie wavelength are two different concepts that describe different aspects of quantum mechanics. The Heisenberg Uncertainty Principle states that there is a fundamental limit to how precisely we can know the position and momentum of a particle at the same time. On the other hand, the de Broglie wavelength describes the wave-like nature of particles and how their momentum is related to their wavelength.

In the case of your homework question, the Heisenberg Uncertainty Principle is used to determine the range of possible velocities for the particle, given its mass and the size of the box it is in. This is because, as you correctly stated, the uncertainty in position (\Delta x) is related to the uncertainty in momentum (\Delta p) by the equation \Delta x \Delta p \geq \frac{h}{2}. This means that the more precisely we know the position of the particle, the less precisely we can know its momentum, and vice versa.

On the other hand, the de Broglie wavelength describes the quantized energies and velocities of particles. This is because, according to the wave-particle duality of quantum mechanics, particles can also behave like waves. The de Broglie wavelength is related to the momentum of a particle by the equation \lambda = \frac{h}{p}. This means that particles with larger momentums (or velocities) have shorter wavelengths, and vice versa.

So, to reconcile the two formulas, we can think of the Heisenberg Uncertainty Principle as placing a fundamental limit on how precisely we can know the momentum (and therefore the velocity) of a particle, while the de Broglie wavelength describes the quantized energies and velocities that a particle can have. Both of these concepts are fundamental to quantum mechanics and are necessary to understand the behavior of particles at the atomic and subatomic level.

I hope this helps to clarify your understanding of these concepts. Keep asking questions and exploring the fascinating world of quantum mechanics!