Relative energies inside a rectangular box

• kraigandrews
In summary, the conversation discusses the energy levels of a three dimensional rectangular infinite potential well with sides of length L, 2L, and 3L. The energy of the first excited state relative to the ground state is given by E = \frac{\hbar^2\pi^2}{2mL^2}\left(\frac{2^2}{1^2}+\frac{1^2}{2^2}+\frac{1^2}{3^2}\right). The second, third, and fourth states can be found by changing the quantum numbers n1, n2, and n3, while keeping the other two constant. The only constraint on the quantum numbers is that they must be positive
kraigandrews

Homework Statement

Consider a three dimensional rectangular infinite potential well with sides of length L, 2L and 3L.
What is the energy of the first excited state relative to the energy of the ground state?
The second state?
The third?
The fourth?

Homework Equations

E=($\pi$2$\hbar$2)/(2mL2)(n_12/L_12+n_22/L_22+n_32/L_33)

The Attempt at a Solution

so E_grounds state= ($\pi$2$\hbar$2)/(2mL2)(1/L12+(1/4L22)+(1/9L32)) then E_1 would be the same thing except n1,n2,n3 are 2 instead of one. however this doesn't yeild the right answer. So I am not sure what I am doing wrong here, my best idea is that n1, n2,n3 are not all 1 for E_ground and E_1 .

Your expression for the energy should be
$$E = \frac{\hbar^2\pi^2}{2mL^2}\left(\frac{n_1^2}{1^2}+\frac{n_2^2}{2^2}+\frac{n_3^2}{3^2}\right)$$
You either have L2 in the coefficient out front or you have L1, L2, and L3 inside the parentheses, not both.

The only constraint on the quantum numbers ni is that they have to be positive integers. The combination n1=n2=n3=1 gives you the lowest energy, so that's the ground state. What combination will give you the next lowest energy? Note you don't have to change all of them.

ok so for E_1 i changed only n_3 and got the correct answer, now I tried to change either the second and couldn't get the answer so I'm a little confused, because changing only n_3 would give the next lowest level for n=3, i believe

nevermind, i figured it out, but could possibly explain how for the second state it, n_2 in the equation uses n=2, i would think that n would =3 as the next in order. Is it because n=2 would give you the next lowest energy? Thanks

I would first clarify that the energy levels in an infinite potential well are quantized and can only take on certain discrete values. The energy of the ground state is the lowest possible energy level and is always considered to be zero. The energy of the first excited state is typically denoted as E1 and is equal to (π²ħ²)/(2mL²). The energy of the second state (E2) is twice that of the first state, the third state (E3) is three times that of the first state, and so on. This is known as the energy level spacing and is a fundamental property of an infinite potential well. Therefore, the energy of the first excited state relative to the ground state is simply E1/E0 = 1, the energy of the second state relative to the ground state is E2/E0 = 2, the energy of the third state relative to the ground state is E3/E0 = 3, and so on. These relative energies do not depend on the dimensions of the rectangular well.

1. What are the factors that affect the relative energies inside a rectangular box?

The relative energies inside a rectangular box are affected by several factors, including the size of the box, the material it is made of, the temperature, and the presence of any external forces or fields.

2. How do the dimensions of the box impact the relative energies inside?

The dimensions of the box play a significant role in determining the relative energies inside. A larger box will have a larger volume and therefore more space for particles to move, potentially resulting in higher relative energies. On the other hand, a smaller box may limit the movement of particles and result in lower relative energies.

3. Can the material of the box affect the relative energies inside?

Yes, the material of the box can impact the relative energies inside. Different materials have different properties, such as density and thermal conductivity, which can affect the movement and distribution of particles and ultimately impact the relative energies.

4. How does temperature influence the relative energies inside the box?

The temperature of the box is directly related to the kinetic energy of the particles inside. As the temperature increases, the particles will move faster and have higher kinetic energies, resulting in higher relative energies inside the box.

5. What role do external forces or fields play in determining relative energies inside a rectangular box?

External forces or fields, such as gravity or magnetic fields, can impact the distribution and movement of particles inside the box, which in turn can affect the relative energies. For example, a strong magnetic field may cause particles to align in a certain way, resulting in a uniform distribution of energies inside the box.

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