Are exp(ikx) and exp(-ax^2) Eigenfunctions of P^2?

In summary, the conversation discusses a question related to position and momentum operators in the context of QM finals past papers. The conversation includes a discussion about the explicit forms of X^2 and P^2 and the eigenfunctions of P^2, with a focus on verifying the answers and proving them.
  • #1
jonnylane
I'm doing some past papers for my QM finals and I've come across a question that is a bit strange. I'm not sure if it's as easy as it sounds.

X and P are one dimensional position and momentum operators, which take the explicit forms of x and -ihd/dx.

i) write down the explicit forms of X^2 and P^2

now then, is this just x^2 and hd^2/dx^2?

im ok on the next few bits, but:

iv) which, if any, of the two functions exp(ikx) and exp(-ax^2) are eigenfunctions of P^2?

my guess is that, since the eigenfunction of P is exp(ikx), its the other one (and the i has disapeared in the squaring process), but how can i prove this?


Im probably just being paranoid, but can someone verify these answers?

thanks
 
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  • #2
Originally posted by jonnylane
now then, is this just x^2 and hd^2/dx^2?

How did you obtain them? that's the important part. I think you're missing a sign on P^2.

iv) which, if any, of the two functions exp(ikx) and exp(-ax^2) are eigenfunctions of P^2?
how can i prove this?

Remember what an eigenfunction is. You need to plug in both functions on P^2, and see if you get a constant times the original function.
 
  • #3


Hi there,

Yes, you are correct in your guess that the eigenfunction of P is exp(ikx) and that the other function, exp(-ax^2), is not an eigenfunction of P^2.

To prove this, we can use the definition of an eigenfunction, which is a function that when operated on by an operator, results in a constant multiple of itself. In this case, we are looking for a function f(x) such that P^2f(x) = cf(x), where c is a constant.

For the function exp(ikx), we have P^2(exp(ikx)) = -k^2(exp(ikx)). Therefore, exp(ikx) is an eigenfunction of P^2 with eigenvalue -k^2.

However, for the function exp(-ax^2), we have P^2(exp(-ax^2)) = -2a(exp(-ax^2) + 2ax^2(exp(-ax^2)). This is not a constant multiple of exp(-ax^2), so it is not an eigenfunction of P^2.

I hope this helps! Good luck on your QM finals.
 

FAQ: Are exp(ikx) and exp(-ax^2) Eigenfunctions of P^2?

What are operators in quantum mechanics?

Operators in quantum mechanics are mathematical entities that represent physical observables in a quantum mechanical system. They can be used to calculate the probabilities of a particle's position, momentum, energy, and other properties.

How do operators act on quantum states?

Operators act on quantum states by transforming them into different states. This transformation can be represented by a mathematical operation, such as multiplication or differentiation, depending on the specific operator and the property it represents.

What is the role of operators in the Schrödinger equation?

The Schrödinger equation, which describes the time evolution of a quantum system, includes operators that represent the system's energy and other physical properties. These operators act on the wave function of the system to determine its evolution over time.

Can operators have eigenstates and eigenvalues?

Yes, operators in quantum mechanics can have eigenstates and corresponding eigenvalues. Eigenstates are quantum states that are unchanged when acted upon by a particular operator, and eigenvalues are the corresponding values that are obtained when measuring the observable represented by the operator.

How are operators related to the Heisenberg uncertainty principle?

The Heisenberg uncertainty principle states that it is impossible to know the exact values of certain pairs of physical properties, such as position and momentum, simultaneously. Operators in quantum mechanics represent these properties and their corresponding uncertainties, and the commutation relations between operators are related to the uncertainty principle.

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