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This tag is for questions relating to Hilbert Space, a vector space equipped with an inner product, an operation that allows defining lengths and angles, and the space is complete. It arises naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces having the property that it is complete. Applies also to pre-Hilbert spaces, rigged Hilbert spaces, and spaces with negative norm or zero-norm states.

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Where is the degeneracy of the bands in this band structure and what does it mean?

In the absence of a lattice potential, the energy eigenstates are simply the momentum eigenstates. The lattice potential can couple together momentum states differing by a momentum $\Delta p = h / L$ …
creillyucla's user avatar
3 votes
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How do you define the "natural" inner product on a tensor product space?

A basis for a tensor product space are the vectors $|v_i\rangle\otimes |w_j\rangle$, where $ i = 1,\ldots,N_V $ and $ j = 1, \ldots, N_W$, where $N_V$ and $N_W$ are the dimensions of $V$ and $W$, resp …
creillyucla's user avatar
7 votes

Understanding the parabolic state of a quantum particle in the infinite square well

There's a simple error in your $\psi_n$ which is screwing you up. In general, the form of the eigenstates $\psi_n(x;x_o,L)$ for a well of length $L$ starting at $x_o$ is $$ \psi_n(x;x_o,L) \propto \si …
creillyucla's user avatar