Why Are Infinite Square Well Eigenstates Not Energy or Momentum Eigenstates?

In summary, the eigenstates of the infinite square well are not energy eigenstates and are not momentum eigenstates. This may seem contradictory at first, but in quantum mechanics, operators do not always behave in the same way as regular numbers. In this case, the square well energy eigenstates are not eigenstates of the momentum operator, but rather of the momentum squared operator. This is due to the fact that the momentum operator is not the "square root" of the momentum squared operator, similar to how -2 is not the square root of (-2)^2.
  • #1
phrygian
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Homework Statement



The eigenstates of the infinite square well are not energy eigenstates and are not momentum eigenstates.

Homework Equations





The Attempt at a Solution



I don't understand how this can be? If the eigenstates of the infinite square well are energy eigenstates, and thus have a definite energy, how can that not have a definite momentum given that momentum in the well is just Sqrt[(2*m*KE)]?
 
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  • #2
What eigenstates are you talking about? i.e. the eigenstates of which operator? There's no such thing as "eigenstates of the infinite square well", there are only eigenstates of each individual operator in the infinite square well potential.
 
  • #3
I am talking about the energy eigenstates of the infinite square well, how are these not also momentum eigenstates since p = Sqrt[(2*m*KE)]
 
  • #4
phrygian said:
I am talking about the energy eigenstates of the infinite square well, how are these not also momentum eigenstates since p = Sqrt[(2*m*KE)]
It's not that simple in quantum mechanics. The formula you have really only works well for regular numbers. But when you're dealing with operators (as you are in QM), you run into problems with multiple operators squaring to the same thing. It's like how you can have two different numbers that have the same square, one positive and one negative, but more complex, because there can be a much larger number of operators with the same square, and only one of them can really be considered the "square root" of the original operator.

You can actually check this for yourself: take the square well energy eigenfunctions
[tex]\psi(x) = A \sin(k_n x)[/tex]
and first apply the momentum operator to them:
[tex]\hat{p}\psi = -i\hbar\frac{d}{d x}\psi = \cdots[/tex]
Notice that what you get is not a multiple of the original wavefunction, so it's not an eigenfunction. But if you apply the momentum operator again (try it!), you'll get something that is a multiple of the original eigenfunction. So the square well energy eigenstates are eigenstates of [tex]\hat{p}^2[/tex] (because that's what is in the Hamiltonian), but not of [tex]\hat{p}[/tex]. This can happen because [tex]\hat{p}[/tex] is not the operator that you'd call the "square root" of [tex]\hat{p}^2[/tex]:
[tex]\hat{p}\neq \sqrt{\hat{p}^2}[/tex]
kind of like how [itex]-2 \neq \sqrt{(-2)^2}[/itex].
 
  • #5


There are a few key concepts that need to be clarified in order to understand why the eigenstates of the infinite square well are not energy or momentum eigenstates.

Firstly, the eigenstates of a system are the states in which the system's properties are well-defined and unchanging. In the case of the infinite square well, the eigenstates are the stationary states of the system, meaning they do not change with time.

Secondly, energy and momentum are both conserved quantities in a system. This means that they can only take on certain discrete values, rather than being continuous. In the case of the infinite square well, the energy levels are quantized, meaning they can only take on certain discrete values, and the same applies for momentum.

Now, coming to the question of energy and momentum eigenstates, these are states in which the energy or momentum of the system is well-defined and unchanging. In the case of the infinite square well, the eigenstates are not energy or momentum eigenstates because they do not have a definite energy or momentum. Instead, they are a superposition of different energy and momentum states.

To understand this, imagine a particle in the infinite square well. The particle can have different energy and momentum states, and the probability of finding the particle in each of these states is determined by the eigenstate of the system. However, the particle does not have a definite energy or momentum, as it is in a superposition of different states.

In conclusion, the eigenstates of the infinite square well are not energy or momentum eigenstates because they do not have a definite energy or momentum, but rather are a superposition of different states. This concept is a fundamental aspect of quantum mechanics and is necessary to understand the behavior of particles at the microscopic level.
 

FAQ: Why Are Infinite Square Well Eigenstates Not Energy or Momentum Eigenstates?

What is an infinite square well potential?

An infinite square well potential is a theoretical model used in quantum mechanics to describe a particle trapped inside a box with infinite potential walls on all sides. This means that the particle cannot escape from the box, and its energy is limited to discrete values.

What are eigenstates in an infinite square well potential?

Eigenstates in an infinite square well potential refer to the allowed energy levels that a particle can have inside the box. These energy levels are determined by the boundary conditions set by the infinite potential walls and are represented by wave functions that describe the probability of finding the particle at a specific position inside the box.

How are eigenstates calculated in an infinite square well potential?

The eigenstates in an infinite square well potential are calculated by solving the Schrödinger equation, which is a mathematical equation that describes the behavior of quantum particles. The boundary conditions of the infinite potential walls are then applied to the solution, resulting in a set of discrete energy levels.

What is the significance of infinite square well eigenstates?

Infinite square well eigenstates have significant implications in quantum mechanics as they provide a simple model for understanding the behavior of particles in confined spaces. They also demonstrate the quantization of energy in quantum systems, where energy can only take on specific values instead of a continuous range.

How do infinite square well eigenstates relate to the Heisenberg uncertainty principle?

The Heisenberg uncertainty principle states that it is impossible to know the exact position and momentum of a particle simultaneously. In an infinite square well potential, the eigenstates have a well-defined position but a range of possible momentum values, demonstrating the uncertainty principle in quantum systems.

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