Why is momentum the fourier transform of the wavefunction ?

[DoobleD]

I think this is probably a very basic question: why does the Fourier transform of a wavefunction describing position probabilities gives us a function describing momentum probabilities ?

Is there a fairly simple explanation for this ? What leads us to this relation ?

 

[vanhees71]

Everything follows from the Heisenberg algebra. For one-dimensional motion it reads
$$[\hat{x},\hat{p}]=\mathrm{i},$$
where I set $\hbar=1$ for convenience. From that you can derive that in the position representation
$$\hat{p} \psi(x)=\langle x|\hat{p} \psi \rangle=-\mathrm{i} \partial_x \psi(x)=-\mathrm{i} \langle x|\psi \rangle.$$
The generalized momentum eigenstates are definied by
$$\hat{p} u_{p}(x)=p u_p(x).$$
Solving this equations for the eigenstates leads to
$$u_{p}(x)=N \exp(\mathrm{i} p x).$$
The normalization is most conveniently chosen such that
$$\langle p|p' \rangle=\int_{\mathbb{R}} \mathrm{d} x u_p^*(x) u_{p'}(x) = |N|^2 \int_{\mathbb{R}} \mathrm{d} p \exp[\mathrm{i}(p'-p)x] = (2 \pi)^3 |N|^2 \delta(p-p') \stackrel{!}{=} \delta(p-p') \; \Rightarrow \; N=\frac{1}{\sqrt{2 \pi}}.$$
Thus we have
$$u_p(x)=\langle x|p \rangle=\frac{1}{\sqrt{2 \pi}} \exp(\mathrm{i} p x).$$
From this you finally get for the wave function in momentum representation
$$\tilde{\psi}(p)=\langle p|\psi \rangle = \int_{\mathbb{R}} \mathrm{d} x \langle p|x \rangle \langle x |\psi \rangle = \int_{\mathbb{R}} \mathrm{d} x u_p^*(x) \psi(x) = \int_{\mathbb{R}} \mathrm{d} x \frac{1}{\sqrt{2 \pi}} \exp(-\mathrm{i} p x) \psi(x),$$
which is nothing else than the Fourier transformation of the wave function in position representation. In the same way you immediately get the inverse transformation
$$\psi(x)=\int_{\mathbb{R}} \mathrm{d} p u_p(x) \tilde{\psi}(p) = \int_{\mathbb{R}} \mathrm{d} p \frac{1}{\sqrt{2 \pi}} \exp(\mathrm{i} p x) \tilde{\psi}(p).$$

 

[DoobleD]

Thank you for answering !

This provided some insight. However my knowledge of QM and the associated maths is very incomplete and quite shaky. I think I'll come back to your answer later, it'll probably be much clearer then.

There's just one part I wonder about. This :

The generalized momentum eigenstates are definied by

glish ^pup(x)=pup(x).​

Here is what I read in plain english : "There exist a momentum operator and an associated eigenfunction. The eigenvalue is the momentum." If that's a correct reading, then why do we know there must exist a momentum operator, and its eigenfunction ? Hope the question makes sense. I wonder why someone would come with that idea.

Also in the mean time I have read the first chapter of Griffiths's textbook "Introduction to Quantum Mechanics (2nd Edition)". To introduce momentum, he takes the time derivative of the expected value of position, and using Schrödinger equation, he ends up with an expression for the expected value of momentum. Which does look like a Fourier transform. He seems to say this isn't really good "theoretical foundation" though, I suppose it serves more as an intro.

 

[vanhees71]

Yes, that's a very heuristic "derivation". Anyway, as it comes to the foundations and its subtleties, from what I read in this forum, Griffiths's textbook doesn't get the foundations very clear, and this causes a lot of misunderstandings and troubles for the uninitiated reader, which unfortunately is the adressee of this textbook ;-)). I recommend to read a book which is more careful in explaining the foundations. My first QM textbook at the university has been J. J. Sakurai, Modern Quantum Mechanics (2nd edition). I think that's a very good textbook. To get the math right, I strongly recommend Ballentine, Quantum Mechanics - a modern development, because it introduces the rigged-Hilbert space formalism without dwelling too much on a rigorous mathematical treatment.

 

[DoobleD]

Thank you for the recommandations, I'll have a look at those !

 

[Schneibster]

In plain English, the Fourier transform of the wavefunction in position gives the momentum because position and momentum are conjugate under uncertainty. The Fourier transform of the wavefunction of one therefore gives the wavefunction of the other. This is the underlying meaning of the math @vanhees71 explained above.

The underlying physical reason for this is the properties of the vacuum (of spacetime, specifically) and the complementarity of conservation of momentum and continuous symmetry of physics over position under Noether's Theorem. Position and momentum are intimately related by the nature of spacetime itself, and this leads to complementary Heisenberg uncertainty of position and momentum as well as the fact about the Fourier transforms of their wavefunctions that you are trying to understand.

 

[DoobleD]

Thanks, this adds some insight. At least it does illustrate a profound link between position and momentum in general. I'll come back to this question after I get more practice with QM and hopefully this will be clearer.