D'Alembertian in Sakurai's Advanced QM book

In summary: It is old notation for special relativity. It doesn't carry over (as far as I know) into general relativity.
  • #1
Wrichik Basu
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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?

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  • #2
One of the properties of ##\gamma## matrices is $$\left\{\gamma^\mu,\gamma^\nu\right\}=2g^{\mu\nu}$$
 
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  • #3
Gaussian97 said:
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.)
 
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  • #4
Well, this is possible by defining 4-vectors as ##x^\mu=(ct,ix,iy,iz)##. Then $$x^\mu x_\mu=\delta_{\mu\nu}x^{\mu}x^{\nu}=c^2t^2-x^2-y^2-z^2$$
 
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  • #5
Gaussian97 said:
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 learned till date ##x^\mu=(ct,x,y,z)##. I am really getting confused with the notation in this book.
 
  • #6
Wrichik Basu said:
OK, but that's not what most other books use. At least I have learned 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
Gaussian97 said:
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
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.
 
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  • #9
Wrichik Basu said:
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.
 
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  • #10
Wrichik Basu said:
OK, but that's not what most other books use. At least I have learned 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.
 
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  • #11
Advanced Quantum Mechanics is a bit old-fashioned. One reason is the use of imaginary components to describe four-vectors.
 
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Related to D'Alembertian in Sakurai's Advanced QM book

1. What is the D'Alembertian operator in Sakurai's Advanced QM book?

The D'Alembertian operator, denoted by ∇², is a differential operator used in quantum mechanics to describe the behavior of a particle in a potential field. It is defined as the sum of the Laplacian operator and the inverse of the speed of light squared (c^-2).

2. How is the D'Alembertian operator used in quantum mechanics?

In quantum mechanics, the D'Alembertian operator is used to describe the wave function of a particle in a potential field. It appears in the Schrödinger equation, which is the fundamental equation of quantum mechanics, and is used to calculate the probability of finding a particle at a certain position and time.

3. What is the significance of the D'Alembertian operator in Sakurai's Advanced QM book?

The D'Alembertian operator is significant in Sakurai's Advanced QM book because it is a central tool in solving problems related to quantum mechanics. It allows for the calculation of the wave function of a particle in a potential field, which is essential in understanding the behavior of particles at the quantum level.

4. How does the D'Alembertian operator relate to other operators in quantum mechanics?

The D'Alembertian operator is related to other operators in quantum mechanics, such as the Hamiltonian operator and the momentum operator. It can be expressed in terms of these operators, making it a useful tool in solving problems related to quantum mechanics.

5. Are there any limitations to the use of the D'Alembertian operator in Sakurai's Advanced QM book?

While the D'Alembertian operator is a powerful tool in quantum mechanics, it does have some limitations. It can only be used to describe particles in potential fields and does not take into account other factors such as spin or interactions with other particles. Additionally, it is only applicable in non-relativistic situations and cannot be used to describe particles moving at speeds close to the speed of light.

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