- #1
quasar_4
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I need someone to tell me if I'm understanding things. :shy:
Let's say that we're studying the infinite square well problem, where the well extends from -L/2 to L/2 in 1 dimension. In this case, the energy of the system, E, is less than the potential at the barriers, so the eigenstates of the Hamiltonian (obviously) correspond to bound states.
Here is where I am confused - please tell me what I am thinking correctly and incorrectly:
- the Hamiltonian and momentum operators commute, so in general, they share a set of eigenfunctions. But the particle in this well can't be in an eigenstate of momentum, because it's in a bound state (and eigenstates of momentum correspond to scattering problems)?
- We know that the expectation value of momentum, <p>, must be zero for a particle in the well because bound states are stationary states, and a nonzero <p> would indicate that the particle was escaping the well (is this a good sort of physical reasoning)?
- The probability of finding the particle is greater at the center of the well then at the edges , but I can't really explain this physically (it seems to be more a mathematical result in my mind than a physical one, and I'm not sure how to describe the probability of finding the particle at some point, without thinking of probability density functions).
Let's say that we're studying the infinite square well problem, where the well extends from -L/2 to L/2 in 1 dimension. In this case, the energy of the system, E, is less than the potential at the barriers, so the eigenstates of the Hamiltonian (obviously) correspond to bound states.
Here is where I am confused - please tell me what I am thinking correctly and incorrectly:
- the Hamiltonian and momentum operators commute, so in general, they share a set of eigenfunctions. But the particle in this well can't be in an eigenstate of momentum, because it's in a bound state (and eigenstates of momentum correspond to scattering problems)?
- We know that the expectation value of momentum, <p>, must be zero for a particle in the well because bound states are stationary states, and a nonzero <p> would indicate that the particle was escaping the well (is this a good sort of physical reasoning)?
- The probability of finding the particle is greater at the center of the well then at the edges , but I can't really explain this physically (it seems to be more a mathematical result in my mind than a physical one, and I'm not sure how to describe the probability of finding the particle at some point, without thinking of probability density functions).