What Is the Cardinality of Bloch Waves in Quantum Mechanics?

[jostpuur]

According to the Bloch's theorem, the solutions of SE in a periodic potential may be written as superpositions of Bloch waves. But what kind of superpositions are these? There is the continuous wave vector parameter, over which we can integrate just like in forming free wave packets, but what about the cardinality of solutions for some fixed wave vector. Are the wave equations for fixed wave vector always like this

$$\psi(x) = \sum_j u_j(x)e^{ikx},$$

where $u_j(x)$ is some sequence (or finite amount) of periodic functions, or could they be like this

$$\psi(x) = \int d\alpha\; u(x,\alpha) e^{ikx}$$

where we have periodic functions $x\mapsto u(x,\alpha)$ for each $\alpha$, where $\alpha$ is a continuous parameter?

EDIT: Now I see that this is a very strange question, because those superpositions of periodic functions (with same periodicity) are still periodic. I'm not sure what I was thinking... I think I was thinking about how to write superpositions in respect to the wave vector k, but then changed the question while typing this. Can there even be several periodic parts u(x) for some fixed wave vector? Perhaps I should start writing these questions on paper before posting them?

 

[jostpuur]

The wave vector k is not directly related to the energy with Bloch waves. How many different Bloch waves

$$\psi_j(x) = u_j(x) e^{ik_j x}$$

can there be corresponding to the same energy? Could there be eigenstates of the Hamiltonian which have the form

$$\psi(x) = \int d\alpha\; \psi_{\alpha}(x) = \int d\alpha\; u(x,\alpha) e^{ik(\alpha) x}?$$

 

[denian]

The cardinality of Bloch waves refers to the number of unique solutions that can be formed using the Bloch's theorem. These solutions are superpositions of Bloch waves, which are characterized by a continuous wave vector parameter. This means that for each fixed wave vector, there is an infinite number of possible solutions, each corresponding to a different choice of the continuous parameter.

In the first equation provided, the solutions are written as a sum over a finite or countably infinite number of periodic functions, which are multiplied by the exponential term. This is a valid representation of Bloch waves, as each term in the sum represents a different solution.

In the second equation, the solutions are written as an integral over a continuous parameter, with each term corresponding to a different periodic function. This is also a valid representation, as the integral captures all possible solutions for a fixed wave vector.

It is important to note that in both cases, the solutions are still periodic, as the exponential term ensures that the overall wave function remains periodic. Therefore, there can be multiple periodic parts for a fixed wave vector, but they will always combine to form a periodic solution.

In conclusion, the cardinality of Bloch waves is infinite, as there are an infinite number of unique solutions that can be formed using the Bloch's theorem. The specific form of the superposition may vary, but all solutions will be periodic in nature.