Eigenfunctions in Hilbert Space, Infinite Square Wells and Uncertainty

In summary, the conversation discusses questions about quantum mechanics, specifically the ground state of the infinite square well as an eigenfunction of momentum, the uncertainty principle, and the inability for two noncommuting operators to have a complete set of common eigenfunctions. The conversation also touches on issues with using the code tag for formulas.
  • #1
neo2478
8
0
Hi I'm kinda stuck with a couple quantum HW questions and I was wondering if you guys could help.

First, Is the ground state of the infinite square well an eigenfunction of momentum?? If so, why. If not, why not??

Second, Prove the uncertainty principle, relating the uncertainty in position (A=x) to the uncertainty in energy ([tex]B=p^2/(2m + V)[\tex]):

[tex]\sigma x\sigma H \geq \hbar/2m |<P>|[\tex]

For stationary states this doesn't tell you much -- why not??

And finally, Show that two noncommuting operators cannot have a complete set of common eigenfunctions. Hint: Show that if P(operator) and Q(operator) have a complete set of common eigenfunctions, the [P(operator),Q(operator)]f = 0 for any function in Hilbert space.

thanks in advance, Rob.
 
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  • #2
Also can someone tell me why the code thingy for the formulas ain't working??
 
  • #3
neo2478 said:
Also can someone tell me why the code thingy for the formulas ain't working??

Because the end tag of the tex part is [ / tex ] and not [ \ tex ]
 

1. What is an eigenfunction in Hilbert Space?

An eigenfunction in Hilbert Space is a special type of function that remains unchanged when operated on by a linear operator. In other words, it is the solution to an equation where the function and its derivative are proportional to each other.

2. What is an Infinite Square Well?

An Infinite Square Well is a simplified model used in quantum mechanics to represent a particle confined to a finite space. It consists of an infinite potential barrier on either side and a constant potential in between, creating a well-like shape.

3. How are eigenfunctions related to Infinite Square Wells?

In an Infinite Square Well, the allowed energy levels for a particle are represented by the eigenfunctions of the corresponding Hamiltonian operator. These eigenfunctions are also known as stationary states, as the particle's wave function will remain constant over time when in one of these states.

4. What is the Uncertainty Principle?

The Uncertainty Principle, also known as the Heisenberg Uncertainty Principle, states that it is impossible to know both the position and momentum of a particle with absolute certainty at the same time. This is due to the fundamental wave-particle duality of quantum mechanics.

5. How is uncertainty related to eigenfunctions in Hilbert Space?

The Uncertainty Principle is related to eigenfunctions in Hilbert Space through the concept of wave-particle duality. Since the eigenfunctions in Hilbert Space are used to describe the wave-like behavior of particles, the Uncertainty Principle applies to the measurement of these particles. This means that the more precisely we know the position of a particle, the less precisely we can know its momentum, and vice versa.

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