Uncertainty Principle & Probability Density

In summary, the first problem involves an electron passing through a hole and the uncertainty principle predicts that there will be a change in momentum. The uncertainty principle is given by (delta x)(delta px) > h/2, where delta x is the width of the hole. The uncertainty in momentum is estimated by substituting delta x=10 microns or 5x10^-6 m into the formula. The second problem involves finding the distance from the surface of a metal where the electron's probability density is 10% of its value at the surface. The electron's energy in a metal piece is negative because it is bound, and this is compared to the energy of a free electron. The exact values of delta x and delta px are
  • #1
kingwinner
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I find quantum mechanics to be very hard, and I am currently having trobule with the following 2 problems, can someone please help me out?

1) A thin solid barrier in the xy-plane has a 10-micrometer-diameter circular hole. An electron traveling in the z-direction with vx=0 m/s passes through the hole. Afterward, is vx still zero? If not, within what range is vx likely to be?

My textbook says that the uncertainty principle is (delta x)(delta px) > h/2
But I am not sure what delta x would be in this situation...and I got completely confused with the x,y,z direction business as well...if vx is zero originally, why would it possibly be different after going through the hole? I don't understand what is going on...


2) A typical electron in a piece of metallic sodium has energy -Eo compared to a free electron, where Eo is the 2.7 eV work function of sodium. At what distance beyond the surface of the metal is the electron's probability density 10% of its value at the surface?
First of all, I don't get what it means by saying that the elctron has energy "-Eo compared to a free electron", what is the energy of a free electron? why is there a negative sign in front of Eo?

Secondly, I don't even know which equation to use, can someone give me some hints on solving this problem? I am totally lost...


Any help is greatly appreciated!:smile:
 
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  • #2
1) Electron diffraction, gives you that you must have some v_x after passage of the hole.

2) Well the electron energy in a metal piece is negative because the electron is bound. Bound states have negative enery.
 
  • #3
1) The electron is moving in the z-direction only and we are given that v_x=0 m/s, so v_x should be 0 m/s always, right? Passing a hole shouldn't affect anything...there is no force acting on it...
 
  • #4
In quantum mechanics particles aren't points. They are represented by a wavefunction that can be extended in space. In squeezing itself through the hole, the uncertainty principle tells you have to expect the possibility there will be a change in momentum.
 
  • #5
So should I substitute delta x=5x10^-6 m into the formula (delta x)(delta px) > h/2 ?
 
  • #6
Sure. Though be warned if you are comparing with a printed answer they might decide to put delta x=10 microns. This is not an exact use of the uncertainty relation since we haven't clearly defined 'delta'. It's only an estimate.
 
  • #7
Yes, the only way I can match the answer is to substitute delta x=10x10^-6 m, why is that? Is it because it may be at one end?
 
  • #8
Essentially, yes. But realize that answer is not 'correct' either. The exact definitions of the deltas in the uncertainty principle are standard deviations of wavefunctions. In these sorts of problems you aren't given exact wavefunctions, you just 'estimate' the width from the parameters of the problem. It's a 'ball park' estimate. You shouldn't expect it to be exact.
 

1. What is the Uncertainty Principle?

The Uncertainty Principle, also known as the Heisenberg Uncertainty Principle, is a fundamental concept in quantum mechanics that states that it is impossible to simultaneously know the exact position and momentum of a particle. In other words, the more accurately we know the position of a particle, the less accurately we can know its momentum, and vice versa.

2. How does the Uncertainty Principle relate to Probability Density?

The Uncertainty Principle is closely related to the concept of Probability Density. In quantum mechanics, probability density is used to describe the likelihood of finding a particle at a specific position. The Uncertainty Principle states that the product of the uncertainty in position and the uncertainty in momentum of a particle must always be greater than or equal to a certain value, known as Planck's constant.

3. Why is the Uncertainty Principle important?

The Uncertainty Principle is important because it sets limits on what we can know about the behavior of particles in the quantum world. It shows that there is inherent uncertainty in the universe, and that the act of observing a particle can actually change its behavior. This principle has significant implications for the study of quantum mechanics and our understanding of the fundamental workings of the universe.

4. Can the Uncertainty Principle be violated?

No, the Uncertainty Principle cannot be violated. It is a fundamental principle of quantum mechanics that has been extensively tested and confirmed through various experiments. While it may seem counterintuitive, the Uncertainty Principle is an essential part of our understanding of the quantum world and is a necessary concept for making accurate predictions about the behavior of particles.

5. How does the Uncertainty Principle affect our everyday lives?

While the effects of the Uncertainty Principle may not be directly observable in our everyday lives, it has numerous practical applications. For example, it is the principle behind the functioning of electron microscopes, which use the uncertainty of electron position to create high-resolution images. The Uncertainty Principle also plays a crucial role in modern technologies such as lasers and transistors, which rely on the behavior of particles at the quantum level.

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