Group of Wave Vector for k - Action of Space Group

[hokhani]

For a specific wave vector, $k$, the group of wave vector is defined as all the space group operations that leave $k$ invariant or turn it into $k+K_m$ where $K_m$ is a reciprocal vector. How the translation parts of the space group, $\tau$, can act on wave vector? Better to say, the dimension of a wave vector is $1/length$ while the translation operator acts on the lengths!

 

[Dr_Nate]

You are using the word space in space group as physical space. You can have space group symmetries without having dimensions of length.

Your question makes me wonder if you know what a wave vector looks like in reciprocal space and what it represents in a wave function.

 

[hokhani]

You are using the word space in space group as physical space. You can have space group symmetries without having dimensions of length.

Your question makes me wonder if you know what a wave vector looks like in reciprocal space and what it represents in a wave function.

Many thanks for your answer.
I don't know about that. Could you please help me with how a translation operator $\tau$ act on a wave vector $k$?

 

[Dr_Nate]

It sounds like you are trying to apply a spatial-translation operator to reciprocal space. There's more than one type of translation operator. For example, there is also a time-translation operator. Instead you want a translation operator specific to reciprocal space.

 

[hokhani]

It sounds like you are trying to apply a spatial-translation operator to reciprocal space. There's more than one type of translation operator. For example, there is also a time-translation operator. Instead you want a translation operator specific to reciprocal space.

My question still remains.

 

[Dr_Nate]

It's real easy. You've pretty much gave it in your original post. Just write out the definition for a spatial-translation operation for a crystal and just substitute the appropriate variables that are in reciprocal space.