Trouble with cubical cavity radiation

[GhostLoveScore]

Hi everyone

I have some trouble understanding quantum physics. Here is a scan from the book from Eisberg and Resnick
Here is how much I understand. This is radiation inside cubical cavity, λ_x/2 is distance from two nodes of component x of the radiation.
Same goes for y and z components. In the picture they've connected the nodes together to make a plane. I understand that.
Next if you could take a look at the equation below, that I marked in red.
2a/λ should be an integer number, isn't it? If that's so, how can we have n_x, n_y, n_z also integers?What is the reason that we didn't choose 2a/λ_x, 2a/λ_y, 2a/λ_z to be integer value instead of 2a/λ?

 

[DrClaude]

2a/λ should be an integer number, isn't it? If that's so, how can we have n_x, n_y, n_z also integers?

The system of coordinates is aligned with the edges of the box. The fact that it is a cubic box with conducting walls results in a quantisation condition in x, y and z, hence _x, n_y, and n_z are integers. For an arbitrary direction, not all values of 2a/λ are allowed, but 2a/λ is not necessarily an integer.

 

[GhostLoveScore]

The system of coordinates is aligned with the edges of the box. The fact that it is a cubic box with conducting walls results in a quantisation condition in x, y and z, hence _x, n_y, and n_z are integers. For an arbitrary direction, not all values of 2a/λ are allowed, but 2a/λ is not necessarily an integer.

OK, I just want to confirm if I understand this correctly.

For some radiation propagating inside the cavity, only its x, y and z components need to satisfy the requirement

2a/λ_x=n_x
2a/λ_y=n_y
2a/λ_z=n_z

where n_x, n_y, n_z is 1, 2, 3...

and then
2a/λ=√(n_x²+n_y²+n_z²)
doesn't have to be integer?Can each x, y, z component have different n (different wavelength)? Or are n_x=n_y= n_z?

 

[DrClaude]

For some radiation propagating inside the cavity, only its x, y and z components need to satisfy the requirement

2a/λ_x=n_x
2a/λ_y=n_y
2a/λ_z=n_z

where n_x, n_y, n_z is 1, 2, 3...

and then
2a/λ=√(n_x²+n_y²+n_z²)
doesn't have to be integer?

Correct.

Can each x, y, z component have different n (different wavelength)? Or are n_x=n_y= n_z?

Any combination is allowed. Note that this restricts the possible direction of propagation of the radiation, as the n's also affect the angles.