Zwiebach Page 172: Understanding Momentum and Length

[ehrenfest]

Homework Statement

I am somewhat confused about why Zwiebach sets up this space as a box with side lengths L1,L2,...,Ld and then requires that

$$p_i L_i = 2 \pi n_i$$

i = 1,2,3,...d

Can somewhat explains what this equation means physically? Why would you multiply momentum by a length? If anything I would think you would divide momentum by a length to get the mass times the time?

And how can we imagine space as a box when I am assuming we are talking about the space in which we live which is infinite in each direction?

Homework Equations

The Attempt at a Solution

 

[nrqed]

Homework Statement

I am somewhat confused about why Zwiebach sets up this space as a box with side lengths L1,L2,...,Ld and then requires that

$$p_i L_i = 2 \pi n_i$$

i = 1,2,3,...d

Can somewhat explains what this equation means physically? Why would you multiply momentum by a length? If anything I would think you would divide momentum by a length to get the mass times the time?

It's just the quantization condition for an infinite square well in quantum mechanics. If you have a well of width "a", and allow the wave to be periodic then the possible values of the de Broglie wavelength are a, a/2, a/3 ...a/n. SO the possible values of p are h/lambda = nh/a = 2 pi n hbar/ a. Setting hbar=1 gives 2 pi n / a.

And how can we imagine space as a box when I am assuming we are talking about the space in which we live which is infinite in each direction?

Homework Equations

The Attempt at a Solution

It's a standard trick of QFT to avoid working with continuous variables and infinities.
He uses periodic conditions so you can imagine that the wavefunction repeats itself over all space and that we are only looking at a sample box of finite volume. Or you can imagine that we will take the volume to infinity at the end of the calculation.

 

[ehrenfest]

I see. This explains why we have to sum over p instead of integrate over it in equations such as 10.58. We are really summing over n_i in equation 10.34 and then summing i from 1 to d, right?

 

[nrqed]

I see. This explains why we have to sum over p instead of integrate over it in equations such as 10.58.

right

We are really summing over n_i in equation 10.34 and then summing i from 1 to d, right?

The notation in that equation is a bit confusing

what he meant to say is that along any of the dimensions, the momentum is quantized in values that depend on the length along that dimension. The momenta really should have two labels: one label to indicate the dimension they correspond to (1,2...d) and the second label to tell us which of the possible mode we are dealing with (n=1,2...infinity).

By the way , this is one example where a repeated index is NOT meant to be summed over!

To be more clear, I think it is better to use letters instead of 1,2...d for the dimensions. So let's say the dimensions are labelled by x,y,z,w... etc. (we would run out of letters quickly but let's use letters for the sake of the explanation).

What Zwiebach means by 10.34 is that, say we work along x, the momenta along x are quantized following

$$(p_x)_{n_x} = \frac{2 \pi n_x}{L_x}$$
where $n_x$ may take any integer value 1,2,3...infinity.

Then there is a corresponding equation along y:
$$(p_y)_{n_y} = \frac{2 \pi n_y}{L_y}$$
and so on.


So the p's really have two labels: one to indicate along which dimension they are (that's the label "i" used by Zwiebach) and then there is another label $n_i$ to indicate which mode we are talking about. But to make things less cluttered, he chose to not write explicitly the label foe the mode since the equation makes it clear that the momentum depends on the mode. A more complete expression would have been
$$(p_i)_{n_i} = \frac{2 \pi n_i}{L_i}$$

I hope this makes sense.

patrick

 

[ehrenfest]

I see. Thanks.