# Infinite potential well and linear superposition

• Vuldoraq
In summary, we discuss the properties of a particle in a potential well with a wave function that is a linear superposition of the lowest two stationary states. We calculated the product of the complex conjugate of the wave function and the wave function itself, which gives a real number. We then simplified the result using Euler's notation and found the frequency of oscillation to be the square root of the sum of the squares of the real and imaginary parts. We also calculated the expectation value of the Hamiltonian operator and found it to be less than the sum of the energy eigenvalues. Finally, we discussed finding the probability for the particle to be in either of the states and identified that it is related to the coefficients of the wave function.
Vuldoraq

## Homework Statement

Consider a particle in a potential ;
$$V(x)=0\ for\ 0<=x<=L$$ and
$$V(x)=\infty\ otherwise$$

. Its wave function is a linear superposition of the lowest two stationary states and given by
$$\psi(x,t)=\frac{1}{2}*(\sqrt{3} \psi_{1}(x,t)-\psi_{2}(x,t)$$)

(a) Use the results fromquestion 2 and ﬁnd ψ(x,t). Then calculate $$|\psi(x,t)|^{2}$$.
Make sure that your result for $$|\psi(x,t)|^{2}$$ is manifestly real (no “i” left in there). What is the frequency of the oscillation in $$|\psi(x,t)|^{2}$$?

(b) Calculate the expectation value of the Hamilton operator ˆH. Compare it with the energy eigenvalues
E1 and E2.

(c) What is the probability for the particle to be in state ψ1 and what is the probability for it to be in the state ψ2?

## Homework Equations

Time dependent wave function for the nth stationary state in this potential well in this case is;
$$\psi_{n}(x,t)=\sqrt{\frac{2}{L}}\sin(\frac{n\pix}{L})\exp{\frac{-iE_{n}t}{h/2\pi}\ n=1,2,3,...$$
and
$$E_{n}=\frac{h^{2}n^{2}}{8mL^{2}}$$

## The Attempt at a Solution

Hi, I can get the solution to the first part of (a), I think,

$$\psi(x,t)=\frac{1}{2}(\sqrt{\frac{6}{L}}\sin(\frac{\pi\x}{L})\exp{\frac{-iE_{1}t}{h/2\pi}-\sqrt{\frac{2}{L}}\sin(\frac{2\pi\x}{L})\exp{\frac{-iE_{2}t}{h/2\pi})$$

Where
$$E_{1}=\frac{h^{2}}{8mL^{2}}$$ and
$$E_{2}=\frac{h^{2}}{2mL^{2}}$$

But after that I have no idea of where to go. When I try and calculate the product of the conjugate times non-conjugate I end up with complex terms that I can't simplify, so my result isn't real. Also I have no idea how you find the frequancy of oscillation in $$|\psi(x,t)|^{2}$$, or how to calculate the expectation of the Hamiltonian operator. Any help would be a godsend to me, I feel completely lost now in QM.

Thanks in advanc

Last edited:
The complex conjugate of (a + ib) is (a - ib). So, the product of these two will give a real number. You stated your result has complex terms. Show how you took the product of the complex number and the conjugate.

The conjugate of this,

$$\psi(x,t)=\frac{1}{2}(\sqrt{\frac{6}{L}}\sin(\frac{\pi*x}{L})\exp{\frac{-iE_{1}t}{h/2\pi}-\sqrt{\frac{2}{L}}\sin(\frac{2\pi*x}{L})\exp{\frac{-iE_{2}t}{h/2\pi})$$

will be,

$$\psi_{1}(x)*\bar{\phi_{1}(t)}-\psi_{2}(x)*\bar{\phi_{2}(t)}$$

where $$\phi_{n}(t)=exp{\frac{-iE_{n}t}{h/2\pi}$$

Now taking $$[\psi_{1}(x)*\bar{\phi_{1}(t)}-\psi_{2}(x)*\bar{\phi_{2}(t)}]*[\psi_{1}(x)*{\phi_{1}(t)}-\psi_{2}(x)*{\phi_{2}(t)}]$$

Gives

$$\psi_{1}(x)^{2}-\psi_{1}(x)*\bar{\phi_{1}(t)}*\psi_{2}(x)\phi_{2}(t)-\psi_{2}(x)*\bar{\phi_{2}(t)}\psi_{1}(x)\phi_{1}(t)+\psi_{2}(x)^{2}$$

since $$\phi_{1}(x)\neq\phi_{2}(x)$$ in general I have two mixed terms in the middle. How do I get rid of them? What have I done wrong?

One way of doing this is to express the compex exponential in terms of cosine and sine using the Euler relation. This will separate the real and imaginary parts then you can combine in the form a + ib. Then multiply by the conjugate giving a2 + b2.

Okay. I'm going to simplify everything and call the arguments of both exponential theta(1) and theta(2).

Expanding using Eulers notation gives,
$$\psi(x,t)=\frac{1}{2}*[\psi_{1}(x)cos{\theta_{1}}-\psi_{2}(x)cos{\theta_{2}+i(\psi_{2}\sin{\theta_{2}+\psi_{1}sin{\theta_{1})$$

Thus, $$\bar\psi(x,t)*\psi(x,t)$$ is,

$$\frac{1}{4}(x^{2}+y^{2})$$

where $$x=\psi_{1}(x)cos{\theta_{1}}-\psi_{2}(x)cos{\theta_{2}$$
$$y=(\psi_{2}\sin{\theta_{2}+\psi_{1}sin{\theta_{1})$$

Does this look alright now?

Looks fine with the exception that I get a minus sign between the sine terms since both arguments are negative.

Excellent thanks for the help, I will put the minus's in at the end since i always lose the damn things!

Now please could you give me a hint as to how I find the frequency of oscillation in $$\bar\psi(x,t)*\psi(x,t)$$?

For part (b) I got the expectation of the Hamiltonian to be less than the sum of the energy eigenvalues. Is this right or have a made a mistake somewhere?

Also for part (c) how do I find the probability for the particle to be in either of the states?

## 1. What is an infinite potential well?

An infinite potential well is a theoretical concept in quantum mechanics that describes a system in which a particle is confined to a finite region with infinite potential energy barriers at the boundaries. This creates a "well" that the particle can move within, but it cannot escape.

## 2. How does the infinite potential well model work?

The infinite potential well model works by using the Schrödinger equation to describe the behavior of a particle within the well. The equation takes into account the particle's kinetic energy and its potential energy, which is zero within the well and infinite outside of it.

## 3. What is linear superposition in the context of an infinite potential well?

Linear superposition is a quantum mechanical principle that states that a physical system can be described by a combination of multiple states, each with its own probability amplitude. In the context of an infinite potential well, this means that a particle can exist in multiple energy states within the well, with each state having a different probability of being observed.

## 4. How does linear superposition relate to the uncertainty principle?

The uncertainty principle states that certain physical properties, such as position and momentum, cannot be simultaneously known with absolute precision. In the context of an infinite potential well, linear superposition allows for a particle to exist in multiple positions within the well, making it impossible to know its exact position at any given time.

## 5. What are the real-world applications of the infinite potential well and linear superposition?

The infinite potential well and linear superposition have been used to explain various quantum phenomena, such as the behavior of electrons in atoms and the interference patterns of particles in double-slit experiments. They also have practical applications in quantum computing and cryptography.

Replies
28
Views
854
Replies
7
Views
562
Replies
4
Views
433
Replies
8
Views
957
Replies
12
Views
626
Replies
14
Views
1K
Replies
3
Views
565
Replies
6
Views
449
Replies
10
Views
1K
Replies
7
Views
3K