Exp(i k.r) ~ A travelling wave ?

[mtrl]

exp(i k.r) ~ A traveling wave ??

I am studying electrons in solids and the wavefunctions of free electrons, which have the form exp(i k.r), are said to be representing traveling waves. Isn't there supposed to be a term involving time in the exponential such as exp (i (k.r -wt)) so that it is a traveling wave?
If exp(i k.r) does not represent a standing wave, then what represents a standing wave?

 

[Dr Lots-o'watts]

r = r(x, y, z, t)

 

[mtrl]

There is a further explanation in the book which i think makes r=r(x,y,z,t) an invalid answer.

The book says that exp(+i k.r) and exp(-i k.r) are traveling waves. And a superposition of these two functions, exp(+i k.r) + exp(-i k.r) = 2 cos(k.r) is a standing wave. If r=r(x,y,z,t), then the superposition is a traveling wave too, isn't it?

 

[nasu]

I am studying electrons in solids and the wavefunctions of free electrons, which have the form exp(i k.r), are said to be representing traveling waves. Isn't there supposed to be a term involving time in the exponential such as exp (i (k.r -wt)) so that it is a traveling wave?
If exp(i k.r) does not represent a standing wave, then what represents a standing wave?

As a general observation, in some books the time factor is left out for convenience but the complete form is assumed to be as you show above.

In quantum mechanics, if you plug in the complete (time included) solution in Schrödinger equation, the temporal part can be separated and then you only look at the time independent Schrödinger equation. The solutions of this last one have only position dependence but in the end you should (if necessary) add the time factor.

 

[mtrl]

Thank you nasu. I have understood it now.
The book on solid state physics by Kittel is really driving me mad. There are almost no explanations during formula derivations and lots of steps are omitted.

 

[Dr Lots-o'watts]

nasu's answer is better. Indeed, r usually doesn't include the time coordinate. My answer needs an unconventional definition for r.

 

[nasu]

Thank you nasu. I have understood it now.
The book on solid state physics by Kittel is really driving me mad. There are almost no explanations during formula derivations and lots of steps are omitted.

I agree with you. It's a good reference if you already know the stuff but not the best introduction.