Solving Double Square Well Potential - Physical Chemistry Homework

[ywkim880801]

this is one of my pchem class homework...
and I have no idea how to solve this...

compute $$\tau$$ for and electron in the double square well potential of width L = 2 Bohr, and depth $$\bar{V}$$ = 4 and separation R = 3. Use atomic units for your computation...

and in his lecture notes... he gave us (tau) = h/$$\DeltaE$$ .

he said it's just a simple plug-in problem... but i don't know where that $$\DeltaE$$

comes from...

please help me...

pchem sucks...

 

[azntoon]

The energy difference \DeltaE in this problem is the difference between the energies of the two quantum states that are separated by the double square well potential. To calculate this, you can use the equation for the energy of an electron in a one-dimensional box: E_n = n^2 \hbar^2 \pi^2 / (2mL^2)Where n is an integer indicating which quantum state the electron is in. For the lower state, n will be equal to 1, and for the higher state, n will be equal to 2. Thus, the energy difference \DeltaE between these two states is given by: \DeltaE = E_2 - E_1 = \hbar^2 \pi^2 / (2mL^2)Substituting in the given values of L, m, and \hbar, we get:\DeltaE = 4 \pi^2 \hbar^2 / (2 \cdot 2 \cdot (4 \pi \epsilon_0)^2 ) = 4 / (4 \pi \epsilon_0)^2Now that we have \DeltaE, we can plug it into the equation for \tau:\tau = h / \DeltaE = (4 \pi \epsilon_0)^2 / 4 = \pi^2 \epsilon_0^2

 

[Entropia]

Dear student,

I understand that you are struggling with your physical chemistry homework on solving the double square well potential and computing the tau value for an electron in this potential. I can assure you that with some understanding and practice, you will be able to solve this problem successfully.

Firstly, let's discuss the double square well potential. It is a model potential used in quantum mechanics to study the behavior of a particle confined in a potential well with two distinct regions. In this case, the regions are separated by a distance R and have a width of L, and the potential depth is represented by \bar{V}. The potential energy function for this system is given by:

V(x) = \begin{cases} 0, & \text{if } 0 < x < R \\ \bar{V}, & \text{if } R < x < L \end{cases}

Now, to compute the value of tau for an electron in this potential, we need to understand what \DeltaE represents. In quantum mechanics, \DeltaE is the difference between the energy levels of the system. In this case, it is the difference between the energy levels of the electron in the two potential regions.

To find this difference, we can use the Schrödinger equation and solve for the eigenvalues of the system. This will give us the energy levels of the electron in the two potential regions. Then, we can calculate \DeltaE by subtracting these two values.

Once we have \DeltaE, we can simply plug it into the equation tau = h/\DeltaE, where h is the Planck's constant. This will give us the value of tau for an electron in the double square well potential.

I understand that this may seem like a complex process, but with some practice and understanding of the concepts, you will be able to solve this problem. I would recommend going through your lecture notes and textbook to understand the mathematical steps involved in solving the Schrödinger equation and finding the energy levels.

Remember, physical chemistry can be challenging, but with perseverance and effort, you will be able to overcome it. Keep practicing and seeking help when needed. Good luck with your homework!