Find Min Energy of Particle Using Uncertainty Principle

In summary, the conversation discusses a homework problem involving a particle of mass m confined to a one-dimensional region of length a. The uncertainty principle is used to obtain an expression for the minimum energy of the particle, and the value of this energy is calculated for a 1g bead on a 10 cm wire and for an electron in a region of 1 A in length. The equations needed for the problem are the uncertainty in position (deltax) being equal to a, and the uncertainty in momentum (deltap) being equal to h-bar/(2a). The kinetic energy equation (1/2)*momentum*velocity is used, but the uncertainty in energy and time are not helpful in this case. The equation E
  • #1
darussiaman
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Homework Statement



A particle of mass m is confined to a one-dimensional region of length a.
a.) use the uncertainty principle to obtain an expression for the minimum energy of the particle

b.) calculate the value of this energy for a 1g bead on a 10 cm wire, and for an electron in a region of 1 A in length.

Homework Equations



umm, as far as I'm aware, the only thing we're supposed to know is deltax*deltap => h-bar/2 and the energy-time analog of that. This is first of 3 semesters of quantum so we're not doing anything deep here, the needed equations on all the homework problems have so far been just the basic, simple ones so this shouldn't be any different...

The Attempt at a Solution



Well, I'm not sure how to set it up. This looks like it should be an easy question but having trouble for some reason. First, is this problem saying that deltax (uncertainty in position) is equal to a? If so, then the uncertainty in momentum is deltap => h-bar/(2a). OKay, so what? Energy ... well, kinetic energy = (1/2)*momentum*velocity ... so i could multiply both sides by that and would end up with uncertainty in energy on the left side, but velocity could be anything?? The delta_energy*delta time doesn't seem to be helpful either because what is change in time?
 
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  • #2


Energy is momentum squared divided by twice the mass:

E=p^2/2m
 

FAQ: Find Min Energy of Particle Using Uncertainty Principle

What is the uncertainty principle?

The uncertainty principle, also known as Heisenberg's uncertainty principle, is a fundamental principle in quantum mechanics that states that it is impossible to know the exact position and momentum of a particle at the same time. This means that the more precisely we know the position of a particle, the less we know about its momentum, and vice versa.

How is the uncertainty principle related to the minimum energy of a particle?

The uncertainty principle is related to the minimum energy of a particle through the energy-time uncertainty principle. This principle states that the product of the uncertainty in a particle's energy and the uncertainty in the time it takes to measure that energy must be greater than or equal to a specific value. This value is known as Planck's constant divided by 4π.

How can we use the uncertainty principle to find the minimum energy of a particle?

To find the minimum energy of a particle using the uncertainty principle, we can rearrange the energy-time uncertainty principle equation to solve for the minimum energy. This gives us the equation E = h/4πΔt, where E is the minimum energy, h is Planck's constant, and Δt is the uncertainty in time. This equation allows us to calculate the minimum energy of a particle based on the uncertainty in its momentum.

What is the significance of finding the minimum energy of a particle?

Finding the minimum energy of a particle is significant because it allows us to understand the fundamental limitations of our ability to measure a particle's energy and time simultaneously. It also provides important insights into the behavior of quantum systems and helps us to better understand the structure of matter.

Are there any practical applications of the uncertainty principle and finding the minimum energy of a particle?

Yes, there are many practical applications of the uncertainty principle and finding the minimum energy of a particle. For example, it is used in the design of electronic devices such as transistors and microchips. It also plays a crucial role in the development of technologies like MRI machines and atomic clocks. Additionally, the uncertainty principle is essential in fields such as quantum computing, quantum cryptography, and quantum teleportation.

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