De Broglie wavelength for (near) stationary objects.

In summary, the maximal De Broglie wavelength of a 1kg ball confined in a one-dimensional box with a length of 1m occurs when the momentum of the ball is at its minimum, which cannot be precisely determined due to the finite space. Using the uncertainty principle, the minimum momentum can be calculated to be \Delta p = \frac{h}{4\pi}. This results in a wavelength of 12.57m, which is larger than the dimensions of the box. This indicates the possibility of observing wave-effects if the ball is observed for a long enough period of time.
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Homework Statement


I must be missing out on some fundamental part of quantum mechanics, since I'm unable to quite grasp the following thought experiment. A 1kg ball is confined in a onedimensional box with a length of 1m. What is it's maximal De Broglie wavelength?

Homework Equations


[tex]\lambda=h/p[/tex] (De Broglie Hypothesis)
[tex]\Delta x \Delta p \ge \frac{h}{4\pi}[/tex] (Heisenberg uncertainty principle)

The Attempt at a Solution



So the way i see it, it's maximal de broglie wavelength will occur when the momentum of the ball is at it's minimum. The momentum cannot be precisely determined to be zero since the ball is confined in finite space. Since it's [tex]\Delta x = 1\textrm{m}[/tex], the minimum in momentum will occur at [tex]\Delta p = \frac{h}{4\pi}[/tex].

The wavelength at this momentum is :
[tex]\lambda=\frac{h}{h/(4\pi)}} = 12.57m[/tex]

Now this wavelength is larger than the dimensions of the confining space :confused: . Could someone shed some light on this? I'm getting a macroscopically tangible wavelength for an object at a very slow speed. Overall, how should i interpret de broglie wavelengths for objects with an extremely small momentum?
 
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  • #2
I think you're going about the question the wrong way, because the UP gives the uncertainty in p, not the value of p and hence lambda. You want to use the fact that the probability drops to zero outside of this region (assuming the potential goes to infinity there). Also, the UP is [itex]\Delta x \Delta p \geq \hbar/2[/itex] not less than or equal to.
 
  • #3
Tomsk said:
I think you're going about the question the wrong way, because the UP gives the uncertainty in p, not the value of p and hence lambda. You want to use the fact that the probability drops to zero outside of this region (assuming the potential goes to infinity there). Also, the UP is [itex]\Delta x \Delta p \geq \hbar/2[/itex] not less than or equal to.

I thought that using the smallest [tex]\Delta p[/tex] as a value for p would be correct since i want the ball to be as stationary as possible. I can't assume that [tex]p = 0[/tex] since I've got a finite [tex]\Delta x[/tex].

I suppose my problem is that I may not decrease p so that the De Broglie wavelength exceeds the spatial dimensions of the container? Correct?

Anyways, even if the momentum of the particle is large enough to fit the wavelength into the container, the near stationary ball still has a macroscopically large wavelength, which I'm not sure how to interpret. Does it mean that we could observe wave-effects if we viewed the ball long enough?

(fixed the typo with the '<')
 
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1. What is the De Broglie wavelength for near stationary objects?

The De Broglie wavelength for near stationary objects is a concept in quantum mechanics that describes the wavelength of a particle with a relatively low velocity. It is given by the equation λ = h/mv, where h is Planck's constant, m is the mass of the object, and v is its velocity.

2. What is the significance of the De Broglie wavelength for near stationary objects?

The De Broglie wavelength for near stationary objects is significant because it shows that all objects, regardless of their mass, have wave-like properties. This was a groundbreaking discovery in quantum mechanics and helped to further our understanding of the behavior of particles on a microscopic level.

3. How does the De Broglie wavelength for near stationary objects relate to the Heisenberg uncertainty principle?

The De Broglie wavelength for near stationary objects is related to the Heisenberg uncertainty principle, which states that it is impossible to know both the position and momentum of a particle with absolute certainty. This is because the more accurately we know the position of a particle, the less we know about its momentum, and vice versa.

4. Can the De Broglie wavelength for near stationary objects be observed in everyday objects?

Yes, the De Broglie wavelength for near stationary objects can be observed in everyday objects, although it is typically too small to be detected. As an example, the De Broglie wavelength of a baseball moving at 90 miles per hour is approximately 2.5 x 10^-34 meters, which is incredibly tiny.

5. How is the De Broglie wavelength for near stationary objects different from the De Broglie wavelength for moving objects?

The De Broglie wavelength for near stationary objects is different from the De Broglie wavelength for moving objects because it is calculated using a different equation. The De Broglie wavelength for moving objects is given by the equation λ = h/p, where p is the momentum of the object. Additionally, the De Broglie wavelength for moving objects is typically larger than the De Broglie wavelength for near stationary objects, as the momentum of a moving object is greater than that of a near stationary object.

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