# Does additivity apply to Fourier transform of the wave function

• Heimisson
In summary, the conversation discusses the formula \phi(k-a)=\phi(k)-\phi(a) where k=p/h (h bar that is) and a is a constant, and \phi is the Fourier transform of a wave function (momentum function). The speaker mentions having trouble connecting the physics and mathematics in quantum mechanics and asks if eik(x-a) = eikx - eika. They later realize that this is a trivial question and retract it.
Heimisson
I was wondering if this is correct:

$$\phi$$(k-a)=$$\phi$$(k)-$$\phi$$(a)

Where k=p/h (h bar that is) and a is some constant and $$\phi$$ is the Fourier transform of a wave function (momentum function).

I know that if I had some real formula for $$\phi$$ I could just test this but the problem isn't like that.

I fairly recently started studying quantum mechanics so I'm still in that stage of having a bit of trouble connecting the physics and mathematics.

thanks

A typical wavefunction is a plane wave expressed as eikx. Is

eik(x-a) = eikx - eika?

this was a stupid question...never mind

## 1. What is additivity in the context of Fourier transform?

Additivity refers to the property that the Fourier transform of the sum of two functions is equal to the sum of the individual Fourier transforms of each function. In other words, the Fourier transform operator behaves additively.

## 2. Does additivity apply to the Fourier transform of the wave function in quantum mechanics?

Yes, additivity applies to the Fourier transform of the wave function in quantum mechanics. This means that the Fourier transform of the sum of two wave functions is equal to the sum of the individual Fourier transforms of each wave function.

## 3. How does additivity impact the interpretation of the Fourier transform in quantum mechanics?

Additivity is an important property in quantum mechanics as it allows us to easily compute the Fourier transform of a complex wave function by breaking it down into simpler components. It also allows us to interpret the Fourier transform as a superposition of different frequency components.

## 4. Are there any limitations to the additivity of the Fourier transform in quantum mechanics?

While additivity generally holds for the Fourier transform in quantum mechanics, there are some cases where it may not apply. For example, if the wave function is not square-integrable, the Fourier transform may not be well-defined and additivity may not hold.

## 5. How is additivity related to the uncertainty principle in quantum mechanics?

Additivity plays a crucial role in the uncertainty principle in quantum mechanics. The uncertainty principle states that the more precisely we know the position of a particle, the less precisely we can know its momentum, and vice versa. Additivity allows us to compute the Fourier transform of the wave function, which is related to the momentum of the particle, and understand how changes in position and momentum are related.

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