B Understanding Fourier Transform for Wavefunction Representation in K Space

[Physgeek64]

I understand that the Fourier transform to obtain the representation of a wavefunction in k space is

$$ \phi(k) =\frac{1}{2\pi}\int{dx \psi(x)e^{-ikx} } $$
and that $$p=\bar{h} k$$

But why then is $$\phi(p) =\frac{\phi(k)}{\sqrt{\bar{h}}} $$

Many thanks in advance :)

 

[PeterDonis]

But why then is
$$
\phi(p) =\frac{\phi(k)}{\sqrt{\bar{h}}}
$$

Where are you getting this equation from?

 

[dextercioby]

In QM of non-specially relativistic particles, the Fourier transform is usually defined with a $\frac{1}{\sqrt{2\pi\hbar}}$ factor in front of the integral. Thus the OP should be amended by the user, so his (intended) question should automatically find an answer.

 

[Eye_in_the_Sky]

I understand that the Fourier transform to obtain the representation of a wavefunction in k space is

$$ \phi(k) =\frac{1}{2\pi}\int{dx \psi(x)e^{-ikx} } $$
and that $$p=\bar{h} k$$

But why then is $$\phi(p) =\frac{\phi(k)}{\sqrt{\bar{h}}} $$

Well, I think you would agree that we must have

dp |ø(p)|² = dk |ø(k)|² ,

for the corresponding intervals [p, p + dp] and [k, k + dk], where p=h_bark.

But dp = h_bar dk; so,

h_bar dk |ø(p)|² = dk |ø(k)|² ,

and therefore,

|ø(p)|² = |ø(k)|²/h_bar .

 

[vanhees71]

That's simply, because $|\phi(p)|^2$ is a distribution function (namely the probability distribution for momentum). In this sense we can write
$$|\phi(p)|^2 = \frac{\mathrm{d} N}{\mathrm{d} p},$$
where $N$ is the number of particles. Since the wave number and momentum are related by $p=\hbar k$ you get
$$\frac{\mathrm{d} N}{\mathrm{d} p} = \frac{\mathrm{d} N}{\mathrm{d} k} \frac{\mathrm{d} k}{\mathrm{d} p} = \frac{1}{\hbar} \frac{\mathrm{d} N}{\mathrm{d} k}.$$
Now the phase of the wave function is arbitrary, and thus you can conclude from this that
$$\phi(p)=\frac{1}{\sqrt{\hbar}} \tilde{\phi}(k) = \frac{1}{\sqrt{\hbar}} \tilde{\phi}(p/\hbar).$$
Note that you should use a different function symbol for the momentum and the wave-number distribution!