Physics Friday 46

Quantum Mechanics and Momentum
Part 3: The Fourier Transform, Dirac Delta, and Plancherel’s Theorem

Consider the Fourier transform in one dimension. There are several common conventions for defining it; here, we will use the unitary angular form frequently used in physics: for function f(t) the Fourier transform is
,
and the inverse transform is


These convert between the time domain and the angular frequency domain. For transforms on a function of a position variable f(x), we have
,
and

which convert between the position domain and the (angular) wavenumber domain.
For an n dimensional position space with position vector we find the transform of a function to be in the n dimensional wavenumber space with wavenumber vector
via

and


Now, consider the Fourier transform of the Dirac delta “function”:

and by examination of this, and the inverse transform, we obtain
.
(Note that this is not a mathematically rigorous equality, but is valid under integration with a function: .)
This relation is useful for proving a number of relations, one of which follows.

For a valid wavefunction of one spatial dimension ψ(x), let us define its Fourier transform φ(k).
Then

The complex conjugate of this is

where * indicates the complex conjugate.
Thus, let us now find :

This is known as Plancherel’s Theorem (or in some physics and engineering sources as Parseval’s Theorem), and we see that if the (position space) wavefunction ψ(x) is normalized, , then the corresponding “wavenumber space” wavefunction φ(k) is also normalized.

The reason we want to use such a Fourier transformed function goes back to our de Broglie relations, particularly . Using this, we see that our “wavenumber domain,” via this equation, represents the momentum space basis for our quantum state.

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7 Responses to “Physics Friday 46”

  1. Physics Friday 47 « Twisted One 151’s Weblog Says:

    […] we can define a Fourier transform of this to the (angular) wavenumber domain, φ(k): Now, as we noted previously, the relation shows that the wavenumber domain represents momentum space. Now, recall that for an […]

  2. Physics Friday 49 « Twisted One 151’s Weblog Says:

    […] transform applied to a function of time returns a function of (angular) frequency: (introduced here). We also have the deBroglie relation between energy and angular frequency E=ℏω. Thus, […]

  3. Monday Math 100 « Twisted One 151's Weblog Says:

    […] One of the oldest and best known integral transforms is the Laplace transform; it and the related Fourier transform are likely the most commonly used integral transforms. Like the Fourier transform, it transforms a […]

  4. Monday Math 118 « Twisted One 151's Weblog Says:

    […] By twistedone151 The Laplace Transform Part 19: Inverse Laplace Transform Recall from here the Fourier transform and its inverse . Combining these, we see . Now, letting for t<0, the […]

  5. Monday Math 128 « Twisted One 151's Weblog Says:

    […] Math 128 By twistedone151 We have previously discussed the Fourier transform (here and here, especially). In this post, we noted that (using the symmetric angular convention) the […]

  6. Juho Says:

    Cool! But how can one combine ω and k in the Fourier Transform and in what conditions? I would like to “derive” Schödinger equation but I need to justify why to use ψ(x,t) that contains both ω and k. So via expection value calculations Schödinger equation can be “derived”.

    • twistedone151 Says:

      One can have both ω and k if your function depends on both time t and space x, as each of the former is the variable for the Fourier transform of the corresponding latter variable. I touch on the time transform, and its relation to the time-dependent Schrödinger equation in part 6 of this series.

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