Physical Chemistry/Quantum Mechanics Eigenvalues

In summary, the conversation discusses various expressions and their potential to yield eigenvalue equations. The speaker calculates the derivatives of these expressions and concludes that none of them result in the original wave function multiplied by an eigenvalue. However, there is confusion about what operator the functions should be an eigenfunction of. The speaker mentions using the Hamiltonian operator on a wave function to obtain the energy of the system as the "eigenvalue."
  • #1
trf5
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Homework Statement


Indicate which of the following expressions yield eigenvalue equations and identify the eigenvalue.

a) d/dx (sin(∏x/2))
b) -i*hbar * ∂/∂x (sin(∏x/2))
c) ∂/∂x (e-x^2)

The Attempt at a Solution


I know that if the wave equation yields an eigenvalue equation, it will give me the wave equation multiplied by the eigenvalue back. I calculated the derivatives:

a) 1/2∏cos(∏x/2)
b)(-i*hbar∏/2)(cos(∏x/2))
c) -2x

Because none of these give me back the original wave function, does that mean none of these are eigenvalue equations? I don't think my professor would give us a problem that asks to calculate the eigenvalues, where there are no eigenvalues. In parts A and B, the ∏x/2 is in both the question and the answer, but the sin changes to cosine when you take the derivative, so does that mean it is not an eigenfunction? Any help would be appreciated.
 
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  • #2
That seems like an unusual way to phrase the question. Are you really asking whether any of those functions satisfy ##y''+\lambda y = 0##? Note the second derivative, not just the first.
 
  • #3
I don't think so. I have taken a differential equations course and I have not seen eigenvalues in the context he is asking for before. From my notes, using the schroedinger wave equation (time independent), you can use the Hamiltonian operator on a wave function, and if the wave exists, after using the Hamiltonian, you get the energy of the wave multiplied by the wave function. Therefore, I should get the wave function back in my answers and what he is calling the "eigenvalue" is the energy of the system. Is that any clearer?

Edit: Oops, I just realized I posted in the math section.
 
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  • #4
Well, I don't know about that physics stuff, so it may in fact be clearer, but not to me. But sines and cosines do give multiples of themselves back on differentiating twice. For example, this shows up solving the one dimensional wave equation for a vibrating string with fixed ends, where they are the eigenfunctions. That's why I speculated as I did in post #2. But I will leave it to one of the resident physicists to actually answer your question.
 
  • #5
There's no way to say whether a function is an eigenfunction until you say what operator they are supposed to be an eigenfunction of. Every function is eigenfunction of the identity operator. What hamiltonian are you talking about? If the operator part is supposed to be the d/dx and the rest of the expression the eigenvector then I'd agree none of them are eigenvectors. Pretty badly phrased question.
 
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FAQ: Physical Chemistry/Quantum Mechanics Eigenvalues

1. What are eigenvalues in physical chemistry/quantum mechanics?

Eigenvalues are a mathematical concept used in physical chemistry and quantum mechanics to describe the energy levels of a system. They represent the values of energy that a system can have, and are often used to solve equations and predict the behavior of particles.

2. How are eigenvalues calculated in physical chemistry/quantum mechanics?

Eigenvalues are calculated using the Schrödinger equation, which describes the behavior of quantum systems. The equation involves the use of operators, which act on wave functions to give eigenvalues. The process of calculating eigenvalues can be complex and often involves the use of advanced mathematical techniques.

3. Why are eigenvalues important in physical chemistry/quantum mechanics?

Eigenvalues are important because they provide information about the energy levels of a system, which is crucial in understanding the behavior of particles at the quantum level. They are also used to solve equations and make predictions about the behavior of particles, making them a fundamental concept in physical chemistry and quantum mechanics.

4. How do eigenvalues relate to eigenfunctions in physical chemistry/quantum mechanics?

Eigenfunctions are mathematical functions that correspond to eigenvalues in physical chemistry and quantum mechanics. They describe the wave-like behavior of particles and are used to calculate the probability of finding a particle in a particular energy state. Eigenvalues and eigenfunctions are closely related and are used together to understand the behavior of quantum systems.

5. Can eigenvalues have negative values in physical chemistry/quantum mechanics?

Yes, eigenvalues can have negative values in physical chemistry and quantum mechanics. This is because they represent energy levels, and energy can be either positive or negative. Negative eigenvalues are often associated with bound states, where the energy of a particle is confined to a specific region. Positive eigenvalues, on the other hand, are associated with unbound states, where a particle can exist at any energy level.

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