Quantum Mechanics Infinite Potential Well -- Check Answers please

In summary: You need to average the inverse, not to inverse the average.In summary, the conversation discusses a particle of mass m in a 1-D infinite potential well of width a and its corresponding eigen-functions and eigen-energies. The expansion of the wavefunction is given in terms of the eigenfunctions and the expansion coefficients are explicitly computed. The wavefunction at a later arbitrary time is also calculated. The probabilities of measuring certain eigenvalues are discussed, with the probability of measuring E2 being found to be 16/π². The concept of energy eigenvalues being observed is also mentioned and an expression for the expectation value of the inverse energy is given, with the method used to find it being incorrect.
  • #1
Irishdoug
102
16
Homework Statement
I'm self studying so I just want to ensure my answers are correct so I know I truly understand the material as it's easy to trick yourself in thinking you do!

A particle of mass m is in a 1-D infinite potential well of width a given by the potential:

V= ##\sqrt{4]/{a}## for 0 ##\leq## x ##\leq## a/4
= ##\infty## elsewhere
Relevant Equations
See below.
I'm self studying so I just want to ensure my answers are correct so I know I truly understand the material as it's easy to trick yourself in thinking you do!

A particle of mass m is in a 1-D infinite potential well of width a given by the potential:

V= 0 for 0##\leq## x ##\leq## a
= ##\infty## for x<a ; x>a

The particle is located in the leftmost quarter of the well at time t = 0 and is described by the wave-function:

##\Psi (x, t=0)## = ##\sqrt{\frac{4}{a}}## for 0 ##\leq## x ##\leq## a/4
= 0 elsewhere

The eigen-functions and eigen-energies are given by ##\phi##(x) = ##\sqrt{\frac{2}{a}}## sin(##\frac{n\pi x}{a}##) and ##E_{n}## = ##{{\hbar}^2 {\pi}^2 {n}^2}/{2m {a}^2}##

Q.1 Write the expansion of the wavefunction ##\Psi ##(x,0) in terms of the eigenfunctions and explicitly compute the expansion co-efficients ##c_{n}##

##\Psi##(x,0) = ##\sum_{n=1}^{\infty} c_{n} \phi_{n}## = ##c_{1} \sqrt{\frac{2}{a}}## sin(##\frac{\pi x}{a}##) + ##c_{2} \sqrt{\frac{2}{a}}## sin(##\frac{2 \pi x}{a}##)... etc.

##c_{n}## = ##\int_{0}^{a/4} \sqrt{\frac{2}{a}} sin( \frac{n\pi x}{a}) \Psi (x,0)dx ##

##\rightarrow## ##\sqrt{\frac{2}{a}} \sqrt{{4}/{a}}## ##\int_{0}^{a/4} sin( \frac{n\pi x}{a}) dx ##

##\rightarrow## -##\frac{8}{a} ( \frac{a}{n \pi} cos (\frac{n\pi x}{a}##))

##\rightarrow## carry out substitution for x

##\rightarrow## -##\frac{8}{n\pi}( cos (\frac{n\pi}{4})) - 1##

so; ##c_{n}## = ##\frac{8-4\sqrt{2}}{n\pi }## for n = odd ; ##\frac{8}{n\pi }## for n = even

Q.2 Give an expression for ##\Psi##(x,t) for some later arbitrary time t

##\Psi##(x,t) = ##c_{n}## ##\phi_{n}(x)## ##exp(\frac{-i Ent}{\hbar})## = ##c_{n}## ## \sqrt{\frac{2}{a}}## ## sin( \frac{n\pi x}{a}) ## ##exp(\frac{-i Ent}{\hbar}##)

Q.3 If an energy measurement is made, what values will be observed and with what probabilities?

I said multiples of ##E_{0}## i.e ##n^{2}## ##(\frac{{\hbar}^2 {\pi}^2}{2m {a}^2}##) where n= 1, 2, 3 etc.

I'm not sure if this is correct. The probabilities are then given by ##|c_{n}|^{2}##

Q.4. What is the probability of measuring ##E_{2}##?

##{|c_{2}|^2}## = ##{\frac{-8}{2\pi}}^2## ##({cos\frac{2\pi}{4} -1})^2##

##\rightarrow## (##{\frac{-8}{2\pi}})^2##(##{-1}^2##) = ##\frac{16}{\pi \pi}##

Q.5 Are there energy egenvalues that will never be observed?

In theory, all eigenvalues can be observed, however in reality some will not as there is not enough time to make measurements to observe all eigen-values.

**I presume this is incorrect but I am unsure how to answer it**

Q.6 Give an expression for the expectation value of the inverse energy ##<\frac{1}{E}>## as a function of time.

##\int_{0}^{a/4} \phi^{*} E \phi dx##

##\frac{\hbar^{2}}{2m}## ##\frac{\partial^2 sin(\frac{n \pi x}{a} )exp(\frac{iEt}{\hbar})}{\partial x^2}##

= ##\frac{\hbar^{2}}{2m}## (-##\frac{a}{n \pi}^{2} ##) ##sin(\frac{n \pi x}{a} )exp(\frac{-iEt}{\hbar})##

so, -##\frac{4 \hbar^{2}}{2m n^{2} \pi^{2}}## ##\int_{0}^{a/4}## ## sin(\frac{n \pi x}{a} )exp(\frac{iEt}{\hbar}) ## ##sin(\frac{n \pi x}{a} )exp(\frac{-iEt}{\hbar})## dxThe exponentials cancel are left with:

-##\frac{4 \hbar^{2}}{2m n^{2} \pi^{2}}## ##\int_{0}^{a/4}## ## sin^{2}(\frac{n \pi x}{a} )##

= -##\frac{4 \hbar^{2}}{2m n^{2} \pi^{2}}## (##\frac{ -2a sin(\frac{n\pi}{2}) - \pi an}{8\pi n}##)

= +##\frac{4 \hbar^{2}}{2m n^{2} \pi^{2}}## (##\frac{ 2a sin(\frac{n\pi}{2}) - \pi an}{8\pi n}##)

You then put this over 1 to get the answer.

I believe this is incorrect however as I have no time dependency! Can someone point out where I have gone wrong. Thankyou.
 
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  • #2
I think you need to find ##<1/E>## whereas you have found ##<E>##. You can use following:
$$<E^{-1}>= \sum_n|c_n ^2| E_n ^{-1}$$
 
  • #3
What I did was find <E>, and then I put this over 1. My answer should read:

##\frac{2mn^{2}\pi^{2}}{4\hbar^{2}} \frac{8 \pi n}{2a sin(\frac{n\pi}{2}) - \pi n}##

Is this method incorrect?
 
  • #4
Irishdoug said:
What I did was find <E>, and then I put this over 1. My answer should read:

##\frac{2mn^{2}\pi^{2}}{4\hbar^{2}} \frac{8 \pi n}{2a sin(\frac{n\pi}{2}) - \pi n}##

Is this method incorrect?
Yes, it's incorrect. E.g. the average of ##1,3## is ##2##. But the average of ##1, 1/3## is not ##1/2##.
 
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Likes Abhishek11235 and Irishdoug

Related to Quantum Mechanics Infinite Potential Well -- Check Answers please

1. What is a quantum mechanics infinite potential well?

A quantum mechanics infinite potential well is a theoretical concept in quantum mechanics that describes a one-dimensional system in which a particle is confined within an infinitely high potential energy barrier. This means that the particle is restricted to a specific region and cannot escape, similar to a particle trapped in a box.

2. How does a particle behave in an infinite potential well?

In a quantum mechanics infinite potential well, the particle is described by a wave function that represents the probability of finding the particle at a given position. The particle is in a state of superposition, meaning it exists in all possible positions within the well simultaneously. However, when the particle is measured, it will collapse into a specific position within the well.

3. What is the significance of the infinite potential well in quantum mechanics?

The infinite potential well is a fundamental concept in quantum mechanics and is used to explain various phenomena, such as the quantization of energy levels and the wave-particle duality of matter. It also serves as a simple model for more complex systems, such as atoms and molecules.

4. What is the difference between a finite and infinite potential well?

In a finite potential well, the particle is confined within a region with finite potential energy barriers. This means that there is a possibility for the particle to tunnel through the barriers and escape. In contrast, an infinite potential well has infinitely high barriers, making it impossible for the particle to escape.

5. Can a particle have a non-zero probability of being outside an infinite potential well?

No, a particle in an infinite potential well has a wave function that is zero outside the well. This means that the particle has a zero probability of being found outside the well. However, there is a small chance that the particle can tunnel through the barriers, but this probability decreases as the potential well becomes deeper.

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