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Equation of a Plane Wave, confused

  1. Aug 30, 2014 #1
    Plane wave equation:

    $$\psi(t) = \psi_0e^{i(\vec{k}\cdot\vec{r}-\omega t)}$$

    The part that makes the domain of [itex]\psi(t_i)[/itex] a plane is the k dot r part.

    I'm reading a book that takes this term and imposes the following condition:

    $$\vec{k}\cdot\vec{r}=Const.$$

    which, i understand its necessity, but if we just plug in the LHS of the equation, the information on the RHS is lost no? i mean, we didn't use it; we just got rid of it. Can someone clarify this part for me please.
     
  2. jcsd
  3. Aug 30, 2014 #2

    Simon Bridge

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    ##\vec k\cdot\vec r = \text{const.}## would mean that ##\psi## is a function of time alone.
    Isn't a plane wave also a function of space?
    http://en.wikipedia.org/wiki/Plane_wave#Arbitrary_direction

    You don't "plug in" the LHS of that equation - the equation is a definition of what the LHS means. If you already know what ##\psi(t)## is, then what extra information could the RHS possibly supply?
     
  4. Aug 30, 2014 #3
    aha! i get it! thanks
     
  5. Aug 30, 2014 #4

    vanhees71

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    Hm, I don't get it. What's the book intending to derive/demonstrate? Could you quote more details?
     
  6. Aug 30, 2014 #5

    ehild

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    A wave is a travelling disturbance, and the disturbance described by the function ψ depends both on place and time.
    $$\vec{k}\cdot\vec{r}=Const.$$ is the equation of a wavefront, a plane, where the phase of the wave is the same at each point. Consider a wavefront where C=0 at t=0, that is, ##\vec{k}\cdot\vec{r}=0## . The equation represents a plane at the origin that is perpendicular to the wave vector ##\vec k##. At a later time t, the points where the phase is zero are on the plane
    $$\vec{k}\cdot\vec{r}-ωt=0$$ In case ##\vec k ## is parallel with the x axis, ##\vec k =k\hat e_x##, the plane is perpendicular to the x axis and its position is determined by ##k x -ωt=0##, that is, at ##x=ω/k t ##: the wavefront travels in the positive x dirction, with speed ω/k. ω/k is the propagation velocity or phase velocity of the wave.

    ehild
     
  7. Aug 30, 2014 #6

    nrqed

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    What they are saying is NOT that ## \vec{k} \cdot \vec{r} = constant ## everywhere. What they are saying is this: the vector ## \vec{k} ## is a constant, and the vector ## \vec{r} ## can be anything. Now, you pick constant C. Then all the points satisfying the condition ## \vec{k} \cdot \vec{r} = C ## lie on a plane, right? (and not that that plane will be perpendicular to ##\vec{k}##) What we know is that everywhere on that plane the wave function has the same phase at any given instant (fixed t). So all the points on that plane correspond a fixed phase. This is the definition of a plane wave.
     
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