Boundary conditions of the Casimir's effect on Sakurai

Question

On 3rd edition of Sakurai's modern relativistic quantum mechanics, section 7.8.3 when discussing the Casimir effect, we want to write down an expression for the vacuum energy for two metal plates separated by distance $d$:

$$\tag{7.183}E_0(d)=\hbar\sum_{k_x,k_y,n}\omega_k=\hbar c\sqrt{k_x^2+k_y^2+\bigg(\frac{n\pi}{d}\bigg)^2}$$

The book says this follows from the previous equation where we impose periodic boundary conditioin

$$\tag{7.172}\boldsymbol{k}=(k_x,k_y,k_z)=\frac{2\pi}{L}(n_x,n_y,n_z).$$

How did we get 7.183 from 7.172, why is it not $E_0(d)=\hbar\sum_{k_x,k_y,n}\omega_k=\hbar c\sqrt{k_x^2+k_y^2+\big(\frac{2n\pi}{d}\big)^2}$ ?

Later the book says

$$\tag{7.184} E_0(d)=\hbar c\bigg(\frac{L}{\pi}\bigg)^2\int_0^\infty \mathrm dk_x\int_0^\infty \,\mathrm dk_y \sqrt{k_x^2+k_y^2+\big(\frac{n\pi}{d}\big)^2} $$

Why is the integral from $0$ to $\infty$ not $-\infty$ to $\infty$? How did we get $(\frac{L}{\pi})^2$? Shouldn't it be $\big(\frac{2\pi}{L}\big)^2?$

$\textbf{Edit:}$ I see, so using equation 7.172, we have $$E_0(d)=\hbar\sum_{k_x,k_y,n}\omega_k=\sum_n\hbar c\bigg(\frac{L}{2\pi}\bigg)^2\int_{-\infty}^\infty dk_x\int_{-\infty}^\infty dk_y \sqrt{k_x^2+k_y^2+\big(\frac{2n\pi}{d}\big)^2}$$ this is equal to the integral in 7.184.

Answer 1

Regarding the former issue, the book simply integrates twice from $0$ to $+\infty$ both in $k_x$ and $k_y$ and multiplies the result by a factor $2\times 2$, because the integrated function is symmetric under $k_j\to -k_j$.

Notice that, in fact, a factor $1/4$ which arises from $$dn_x dn_y = \frac{L}{2\pi}\frac{L}{2\pi} dk_xdk_y$$ has been cancelled out by the factor $2\times 2$.

Regarding the latter issue, the book is using Dirichlet boundary conditions: the modes vanish at $0$ and $d$ along $z$. Periodic boundary conditions are imposed only along the $x$ and $y$ direction.

I do not have the book but I expect that it uses vanishing boundary conditions on the two surfaces represented by the plates. This condition produces modes labelled on the positive integers only (think of a particle confined in an infinite double well). The boundary conditions in the directions $x$ and $y$ are not very important since we are considering the limit of infinite plates.