Wave function at high symmetry point

In summary, the conversation discusses the proof of the wave function at the \Gamma point being a real function. It is not true for general k points, but for high symmetry points like X, it can be proven with the assumption of time reversal symmetry. However, if spin orbit coupling is taken into account, time reversal no longer guarantees real valuedness. The conversation also touches on the use of complex exponential vs sine/cosine in choosing periodic functions and the degeneracy of Bloch states in H and translation T_R.
  • #1
jianglai
2
0
How to prove that wave function at [itex]\Gamma[/itex] point can always be a real function? I know it is not true for general k point, but for [itex]\Gamma[/itex] and other high symmetry point like X, is there a simple proof?

Thanks!
 
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  • #2
This is hard to prove as it is wrong in general. E.g. once spin orbit coupling cannot be neglected, the orbitals have to be chosen complex.
 
  • #3
Hmm. If we ignore spin-orbit then this seems easy. Note that the complex conjugate of the Bloch wave at gamma is also a solution of the Schrodinger equation. That means the real and imaginary parts are separately solutions. A similar argument should work at other high-symmetry points if -k = k + K where K is a reciprocal lattice vector.
 
  • #4
A clean discussion involves the assumption and discussion of time reversal symmetry. If there are no spin orbit coupling effects, time reversal will be represented by complex conjugation and the single particle wavefunctions in a periodic potential can always be chosen real as then E(k)=E(-k) so that instead of the solutions [itex]\psi_k(x)=u_k(x)\exp(ikx)[/itex] and [itex]\psi_{-k}=(\psi_k(x))^*[/itex] real valued combinations can be chosen. For k=0, only one real function will be obtained.
If spin orbit coupling is taken into account, time reversal is no longer just complex conjugation so that it does not always guarantee real valuedness. This is known as Kramers degeneracy.
 
  • #5
DrDu said:
A clean discussion involves the assumption and discussion of time reversal symmetry. If there are no spin orbit coupling effects, time reversal will be represented by complex conjugation and the single particle wavefunctions in a periodic potential can always be chosen real as then E(k)=E(-k) so that instead of the solutions [itex]\psi_k(x)=u_k(x)\exp(ikx)[/itex] and [itex]\psi_{-k}=(\psi_k(x))^*[/itex] real valued combinations can be chosen. For k=0, only one real function will be obtained.
If spin orbit coupling is taken into account, time reversal is no longer just complex conjugation so that it does not always guarantee real valuedness. This is known as Kramers degeneracy.

That's right, but I think we want to keep our wave-functions as Bloch waves. In other words, we're really aking where in k-space we can choose the periodic function u_k(r) to be real. I guess you could do what you said for all k if you wanted to work with stationary boundary conditions (in opposition to the conventional Born-von Karmen).
 
  • #6
sam_bell said:
That's right, but I think we want to keep our wave-functions as Bloch waves. In other words, we're really aking where in k-space we can choose the periodic function u_k(r) to be real. I guess you could do what you said for all k if you wanted to work with stationary boundary conditions (in opposition to the conventional Born-von Karmen).

I think it also works with Born- von Karman boundary conditions. So basically the only reason why we have to use complex u_k is because we insist on complex exp(ikx) instead of sin(ikx) or cos(ikx).
 
  • #7
Thank you both for the reply! I think I get a sense of it now. Without spin-orbital coupling, for any [itex]k[/itex], [itex] \psi_{nk}(r) [/itex] and [itex]\psi_{nk}(r)^* = \psi_{-nk}(r) [/itex] are degenerate (in [itex]H[/itex]). But Bloch state are simultaneous eigenstates for both [itex]H[/itex] and translation [itex] T_R [/itex], and only at [itex] -k = k + G [/itex] are [itex]\psi_{nk}(r) [/itex] and [itex]\psi_{-nk}(r)[/itex] degenerate in [itex]T_R[/itex] as well, which means we can take a linear combination of them and get rid of the imaginary part. For a general [itex]k[/itex] however, [itex]\psi_{nk}(r)+\psi_{-nk}(r)[/itex] would be a real-valued eiginstate of [itex]H[/itex] that's not a Bloch state.
 
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  • #8
Couldn't have formulated it better!
 

1. What is a wave function at a high symmetry point?

A wave function at a high symmetry point is a mathematical representation of the quantum state of a system at a point in space where the potential energy is at its highest value. This point is often referred to as a "special point" or "symmetry point" because it possesses a specific set of symmetries that make it easier to solve for the wave function.

2. Why is the wave function at a high symmetry point important?

The wave function at a high symmetry point is important because it provides valuable information about the energy and behavior of a system. It can be used to calculate important physical properties such as energy levels, transition probabilities, and electronic structures. Additionally, the symmetries at these points can reveal important information about the underlying physical laws governing the system.

3. How is the wave function at a high symmetry point calculated?

The wave function at a high symmetry point is typically calculated using mathematical methods such as the Schrödinger equation or the density functional theory (DFT) approach. These methods involve solving for the wave function using the known symmetries at the high symmetry point and considering the effects of the surrounding environment on the system.

4. What is the significance of high symmetry points in crystal structures?

In crystal structures, high symmetry points are particularly important because they often correspond to points of high symmetry in the crystal lattice. This means that the properties of the crystal, such as its electronic and optical properties, can be better understood by studying the wave function at these points. Additionally, high symmetry points can help identify the symmetry groups and space groups of a crystal, which are important for understanding its physical properties.

5. How does the wave function at a high symmetry point differ from other points?

The wave function at a high symmetry point differs from other points because it is characterized by a set of symmetries that make it easier to solve for the wave function. These symmetries result in certain mathematical simplifications that can provide insights into the behavior of the system. At other points, the wave function may not exhibit these symmetries and therefore may be more complex to calculate and interpret.

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