Infinite square well centered at the origin

In summary: I think you lost sight of what you were trying to do. You were looking for ALL solutions to the equation:##\psi'' + k^2 \psi =0##With the boundary conditions that ##\psi(L/2) = \psi(-L/2) = 0##
  • #36
Kaguro said:
To get solution for well from 0 to L ,I should make the transformation x →\to x+L/2

You must be careful with this technique. Which ##x## is which?

In fact, you ended up with them the wrong way round. Instead, try:

##x' = x + L/2##

Where ##x'## is the new variable that runs from ##0## to ##L##.

(Note that this means that ##x = x' - L/2##. Which is not what you did.)
 
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  • #37
Also note that phase factors don't play any role. Defining the system to live in ##[0,L]## the boundary value at ##x=0## tells you ##\psi(0)=0##, which leads to the ansatz ##\psi(x)=A \sin(k x)## for the energy eigenfunctions, then the other boundary condition tells you
$$\psi(L)=A \sin(k L)=0,$$
from which you get ##k \in \pi \mathbb{N}/L##. Thus the complete set of eigenfunctions is
$$\psi_n(L)=A \sin \left(\frac{n \pi x}{L} \right).$$
The factors ##(-1)^j## are phase factors which you can simply neglect, because they have no physical significance.
 
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  • #38
vanhees71 said:
The factors (−1)j(-1)^j are phase factors which you can simply neglect, because they have no physical significance.
Yes, after understanding that x = x' - L/2 will be the correct transformation, I tried it again and got everything correct except this ##(-1)^{n/2}##. Now if you say that it carries no significance and can be neglected, I have arrived at the correct answer! Thank you!
 
  • #39
Kaguro said:
Yes, after understanding that x = x' - L/2 will be the correct transformation, I tried it again and got everything correct except this ##(-1)^{n/2}##. Now if you say that it carries no significance and can be neglected, I have arrived at the correct answer! Thank you!
To explain this a bit more.

What you got was a specific case of:

##\psi(x') = \alpha_n \sqrt{\frac{L}{2}} \sin (\frac{n\pi x'}{L})##

Where ##\alpha_n## is a complex number of unit modulus (sometimes called the phase factor). In this case, you had ##\alpha_n = \pm 1##, depending on ##n##.

This is perfectly valid for a set of eigenfunctions. But, you are free to choose the phase factor to be simply ##1## in each case. Hence, you have the simplest set of solutions:

##\psi(x') = \sqrt{\frac{L}{2}} \sin (\frac{n\pi x'}{L})##:

Note that can always do the opposite as well, and add any phase factors you want. For example, we could take your original solution:

Kaguro said:
##\psi (x) =\begin{cases} \sqrt{ \frac{2}{L}}cos(n\pi x/L) & \text{if n is odd} \\ \sqrt{ \frac{2}{L}}sin(n\pi x/L) & \text{if n is even} \end{cases} ##

And, we could replace this with an alternative, equally valid set of eigenfunctions:

##\psi (x) =\begin{cases} i\sqrt{ \frac{2}{L}}cos(n\pi x/L) & \text{if n is odd} \\ (e^{i\theta_n})\sqrt{ \frac{2}{L}}sin(n\pi x/L) & \text{if n is even} \end{cases} ##

This has all the same properties as your original solution. The only different is that we've added a phase factor to every function.

This is an important point, because when you have a set of eigenfunctions it can be very useful to choose a phase factor in a particular way. Often that some arbitrary complex number is real.

In any case, the point to remember is that you did not find the only possible unique set of solutions. You found a set of solutions, unique only up to the phase factor. You are free, therefore, to simplify or complicate your solution by changing the phase factors any way you like.
 
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  • #40
Thanks to you guys I was having exactly same doubt.
 
<H2>What is an infinite square well centered at the origin?</H2><p>An infinite square well centered at the origin is a theoretical model used in quantum mechanics to describe the behavior of a particle confined within a potential well that extends infinitely in all directions from a central point. The potential within the well is constant, and the particle is unable to escape from it.</p><H2>What is the significance of the origin in an infinite square well?</H2><p>The origin in an infinite square well is the central point where the potential well is located. This point serves as the reference point for the potential energy of the particle within the well. It also represents the center of symmetry for the system, as the potential is the same in all directions from the origin.</p><H2>What are the allowed energy levels in an infinite square well centered at the origin?</H2><p>The allowed energy levels in an infinite square well are quantized, meaning they can only take on certain discrete values. These energy levels are determined by the size of the well and are given by the equation En = n²h²/8mL², where n is the quantum number, h is Planck's constant, m is the mass of the particle, and L is the width of the well.</p><H2>What is the wave function for a particle in an infinite square well centered at the origin?</H2><p>The wave function for a particle in an infinite square well is a mathematical representation of the particle's probability of being found at a certain position within the well. It is given by the equation Ψn(x) = √(2/L)sin(nπx/L), where n is the quantum number and L is the width of the well.</p><H2>How does the behavior of a particle in an infinite square well change with increasing energy?</H2><p>As the energy of a particle in an infinite square well increases, the number of nodes in its wave function also increases. This means that the particle is more likely to be found at the edges of the well, rather than at the center. Additionally, the wavelength of the particle's wave function decreases, making it more confined within the well. This behavior is known as the particle-in-a-box effect.</p>

What is an infinite square well centered at the origin?

An infinite square well centered at the origin is a theoretical model used in quantum mechanics to describe the behavior of a particle confined within a potential well that extends infinitely in all directions from a central point. The potential within the well is constant, and the particle is unable to escape from it.

What is the significance of the origin in an infinite square well?

The origin in an infinite square well is the central point where the potential well is located. This point serves as the reference point for the potential energy of the particle within the well. It also represents the center of symmetry for the system, as the potential is the same in all directions from the origin.

What are the allowed energy levels in an infinite square well centered at the origin?

The allowed energy levels in an infinite square well are quantized, meaning they can only take on certain discrete values. These energy levels are determined by the size of the well and are given by the equation En = n²h²/8mL², where n is the quantum number, h is Planck's constant, m is the mass of the particle, and L is the width of the well.

What is the wave function for a particle in an infinite square well centered at the origin?

The wave function for a particle in an infinite square well is a mathematical representation of the particle's probability of being found at a certain position within the well. It is given by the equation Ψn(x) = √(2/L)sin(nπx/L), where n is the quantum number and L is the width of the well.

How does the behavior of a particle in an infinite square well change with increasing energy?

As the energy of a particle in an infinite square well increases, the number of nodes in its wave function also increases. This means that the particle is more likely to be found at the edges of the well, rather than at the center. Additionally, the wavelength of the particle's wave function decreases, making it more confined within the well. This behavior is known as the particle-in-a-box effect.

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