# Definition of a Gaussian wave packet for a Initial State

• Gabriel Maia
In summary, the author defines the initial state ket as the value of <q|Φin> when <q| is taken with the inner product. The correct definition is with <q|Φin>=\phi_{in}(q)|q> and the inner product is not the same as = \phi_{in}(q).
Gabriel Maia
Hi :)

I'm reading a didactic paper and the author defined the initial state ket as

|$\Phi_{in}$> = ${\int}dq\phi_{in}(q)$|q>

where q is a coordinate and

$\phi_{in}(q)$ = <q|$\Phi_{in}$> = exp$\left[\frac{-q^{2}}{4\Delta^{2}}\right]$

I don't know if I'm missing something but isn't this definition a little flawed? I mean if you calculate the inner product of <q| with the first equation, <q|q>=1, sure, but that does not eliminate the integral, right?

I'm thinking the correct definition would be

|$\Phi_{in}$> = $\phi_{in}(q)$|q>

with

$\phi_{in}(q)$ = <q|$\Phi_{in}$> = exp$\left[\frac{-q^{2}}{4\Delta^{2}}\right]$Thank you

Last edited:
Just think of the unitary operator acting on $|Φ_{in}>$
$\hat{1}|Φ_{in}>$
but then, you also know (complete basis) that:
$\hat{1}= \sum_{q} |q><q| \rightarrow \int dq |q><q|$
the arrow is for a continuous q variable...
By that you get everything straightforwardly...

In the same way, you can get:

$<Φ_{in}|Φ_{in}>= \int dq' dq φ^{*}(q')φ(q) <q'|q>$
$<Φ_{in}|Φ_{in}>= \int dq' dq φ^{*}(q')φ(q) δ(q'-q)$
integrating out q' due to the delta function you get:
$<Φ_{in}|Φ_{in}>= \int dq φ^{*}(q)φ(q) =1$

1 person
I see... so

<q|$\Phi_{in}$> = <q|$\hat{I}$|$\Phi_{in}$>

right? But then

<q|$\Phi_{in}$> = <q|$\hat{I}$|$\Phi_{in}$> = ${\int}dq\phi_{in}(q)$

and the integral of $\phi_{in}(q)$ is not the same as = $\phi_{in}(q)$

what is happening here?

Thank you.

you are confusing the brackets... If you have a continuous variable, then you can't generally say that <q|q>=1. This happens only for discrete ones, because the discrete version of dirac's delta function is the delta of kroenicker...
In fact you have to generalize it for continuous variables, and as I used above:
$<q'|q>= δ(q'-q)$
in the same way for discrete variables you had:
$<m|n>=δ_{mn}$

also avoid using the same q everywhere... since you are already taking the bra <q| in <q|Φin> when you wrote the identity operator you should have used a new symbol like |q'><q'| or generally |something else><something else|

1 person
wait for it... wait for it...

I cannot use the very same variable in $\hat{I}={\int}dq|q><q|$ and in the <q| I took the inner product with, right? The identity operator must be something like

$\hat{I}={\int}dq'|q'><q'|$

This will give me (when I take the inner product with <q|) a Dirac delta and save the day.

I mean... this is the rigourous precedure, right?

Thank you.

Thank you very much :) Saved my day!

## 1. What is a Gaussian wave packet?

A Gaussian wave packet is a mathematical concept used to describe the behavior of a quantum particle in motion. It is a wave function that represents the probability amplitude of finding a particle at a certain position and time, and it is characterized by its shape resembling a bell curve.

## 2. How is a Gaussian wave packet defined?

A Gaussian wave packet is defined by the following equation: Ψ(x,t) = exp[-(x-x0)^2/4σ^2 + iPx/ħ] where x is the position of the particle, t is time, x0 is the initial position of the particle, σ is the width of the wave packet, P is the momentum of the particle, and ħ is the reduced Planck's constant.

## 3. What is the significance of the initial state in a Gaussian wave packet?

The initial state in a Gaussian wave packet represents the starting conditions of the particle's motion. It determines the initial position and momentum of the particle, which in turn affects the shape and behavior of the wave packet as it evolves over time.

## 4. How does a Gaussian wave packet change over time?

A Gaussian wave packet will spread out and become more diffuse as time passes, due to the uncertainty principle. This means that the particle's position becomes less certain, while its momentum becomes more certain. The rate of spreading is determined by the width parameter σ in the wave packet equation.

## 5. What are some real-world applications of Gaussian wave packets?

Gaussian wave packets have numerous applications in physics and engineering, particularly in quantum mechanics, optics, and signal processing. They are used to study the behavior of quantum particles, model wave phenomena such as diffraction and interference, and analyze signals in noise reduction and filtering techniques.

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