D'Alembertian in Sakurai's Advanced QM book

  • #1
1,438
1,300

Main Question or Discussion Point

In Sakurai's book "Advanced QM", he writes the Dirac equation (equation 3.31 to be exact) as: $$\left(\gamma _\mu \ \dfrac{\partial}{\partial\ x_\mu} + \frac{m\ c}{\hbar}\right) \ \psi= 0$$ which we can write as $$\left(\gamma _\mu \ \partial ^\mu \ + \frac{m\ c}{\hbar}\right) \ \psi= 0$$

Next, we go to the section 3-3, and subsection "Free particles at rest".

The author multiples the Dirac equation from the left by ##\gamma _\nu \ \partial ^\nu##. We get the form $$\left[ \partial ^\nu \ \partial ^\mu \ \gamma_\nu \ \gamma_\mu \ - \left(\frac{m\ c}{\hbar}\right)^2 \ \right]\psi = 0$$ If we interchange the indices ##\mu## and ##\nu##, and add to the above equation, we get $$\left[ \partial ^\nu \ \partial ^\mu\ \left( \gamma_\nu \ \gamma_\mu + \gamma_\mu \ \gamma_\nu\right) \ - 2 \ \left(\frac{m\ c}{\hbar}\right)^2 \ \right]\psi = 0$$ which reduces to $$\Box \ \psi - \left(\frac{m\ c}{\hbar}\right)^2 \ \psi = 0$$ using the anticommutation relations of the ##\gamma## - matrices.

Till now, I have just quoted from the book (except changing the ##\partial / \partial x_\mu## to ##\partial ^\mu##).

I knew the D'Alembertian operator to be ##\partial ^\mu \ \partial _\mu##, but Sakurai seems to be suggesting that it is ##\partial ^\mu \ \partial ^\nu## instead. The two can be related: $$\partial _\mu \ \partial ^\mu \ = \ g_{\mu \nu} \ \partial ^\nu \ \partial ^\mu\ ,$$ but that's not what the author has written. What am I missing?

Here is a picture of the page:
Screenshot_20190602-193126.png
 
  • Like
Likes atyy

Answers and Replies

  • #2
353
167
One of the properties of ##\gamma## matrices is $$\left\{\gamma^\mu,\gamma^\nu\right\}=2g^{\mu\nu}$$
 
  • Like
Likes vanhees71, atyy and Wrichik Basu
  • #3
1,438
1,300
One of the properties of ##\gamma## matrices is $$\left\{\gamma^\mu,\gamma^\nu\right\}=2g^{\mu\nu}$$
Thank you, that perfectly solves the problem.

I don't know why Sakurai is reluctant to use the metric tensor. He just writes ##\{ \gamma_\mu , \ \gamma_\nu \} = 2 \delta _{\mu \nu}##. In fact, in chapter 1, he says,
Note that we make no distinction between a covariant and a contravariant vector, nor do we define the metric tensor. These complications are absolutely unnecessary in the special theory of relativity. (It is regrettable that many textbook writers do not emphasize this elementary point.)
 
Last edited:
  • #4
353
167
Well, this is possible by defining 4-vectors as ##x^\mu=(ct,ix,iy,iz)##. Then $$x^\mu x_\mu=\delta_{\mu\nu}x^{\mu}x^{\nu}=c^2t^2-x^2-y^2-z^2$$
 
  • Like
Likes atyy
  • #5
1,438
1,300
Well, this is possible by defining 4-vectors as ##x^\mu=(ct,ix,iy,iz)##. Then $$x^\mu x_\mu=\delta_{\mu\nu}x^{\mu}x^{\nu}=c^2t^2-x^2-y^2-z^2$$
OK, but that's not what most other books use. At least I have learnt till date ##x^\mu=(ct,x,y,z)##. I am really getting confused with the notation in this book.
 
  • #6
353
167
OK, but that's not what most other books use. At least I have learnt till date ##x^\mu=(ct,x,y,z)##. I am really getting confused with the notation in this book.
Yes, I think nowadays using ##(t,x,y,z)## with ##c=1## is the most common way to proceed, so maybe you want to change your book... Sakurai is from 1967.
 
  • #7
1,438
1,300
so maybe you want to change your book... Sakurai is from 1967.
Sakurai is quite a recommended book, that's why I started reading it. But I'll not continue with this book any longer.
 
  • #8
353
167
Well, I've never read Sakurai's book, but it's true it has a lot of popularity. You need to think if it's worth for you. Maybe you can start with something else and then return to Sakurai when you think you'll be able to be comfortable with the notation.
 
  • Like
Likes Wrichik Basu
  • #9
George Jones
Staff Emeritus
Science Advisor
Gold Member
7,369
974
Sakurai is quite a recommended book, that's why I started reading it. But I'll not continue with this book any longer.
Sakurai has written (at least) two different books, "Advanced Quantum Mechanics" and "Modern Quantum Mechanics". I have seen both books recommended, but I have seen "Modern Quantum Mechanics" recommended much more often than "Advanced Quantum Mechanics".

"Modern Quantum Mechanics" is probably the text used most often in North America for graduate quantum mechanics courses.
 
  • Like
Likes Wrichik Basu
  • #10
atyy
Science Advisor
13,913
2,183
OK, but that's not what most other books use. At least I have learnt till date ##x^\mu=(ct,x,y,z)##. I am really getting confused with the notation in this book.
It is old notation for special relativity. It doesn't carry over (as far as I know) into general relativity. You can see an example of the change of notation from SR to GR also in 't Hooft's notes http://www.staff.science.uu.nl/~hooft101/lectures/genrel_2013.pdf.
 
  • Like
Likes Wrichik Basu
  • #11
vanhees71
Science Advisor
Insights Author
Gold Member
2019 Award
15,338
6,730
Advanced Quantum Mechanics is a bit old-fashioned. One reason is the use of imaginary components to describe four-vectors.
 
  • Like
Likes Wrichik Basu

Related Threads on D'Alembertian in Sakurai's Advanced QM book

  • Last Post
Replies
7
Views
2K
Replies
0
Views
870
  • Last Post
Replies
13
Views
3K
  • Last Post
Replies
0
Views
1K
  • Last Post
Replies
16
Views
4K
Replies
6
Views
4K
  • Last Post
Replies
1
Views
917
  • Last Post
Replies
7
Views
855
Top