QFT for Beginners: Operators & Their Physical Significance

[Wrichik Basu]

I'm a beginner in QFT, starting out with D. J. Griffiths' book in this topic.

I have a question on the operators used in QM. What are operators? What is the physical significance of operators? I can understand that $\frac {d}{dt}$ to be an operator, but how can there be a total energy operator, a potential energy operator, and what is the meaning of subtracting them to give a Hamiltonian? ($\hat H = \hat T - \hat V$)

 

[NFuller]

The Hamiltonian is the sum of potential and kinetic energy $\hat{H}=\hat{T}+\hat{V}$ not the difference. The Hamiltonian operator can be said to act as the total energy operator. This means that the eigenvalues of the Hamiltonian are the total energies $E_{n}$ corresponding to a given eigenstate $\phi(\mathbf{x})_{n}$ as
$$\hat{H}\phi(\mathbf{x})_{n}=E_{n}\phi(\mathbf{x})_{n}$$.

 

[Wrichik Basu]

The Hamiltonian is the sum of potential and kinetic energy $\hat{H}=\hat{T}+\hat{V}$ not the difference. The Hamiltonian operator can be said to act as the total energy operator. This means that the eigenvalues of the Hamiltonian are the total energies $E_{n}$ corresponding to a given eigenstate $\phi(\mathbf{x})_{n}$ as
$$\hat{H}\phi(\mathbf{x})_{n}=E_{n}\phi(\mathbf{x})_{n}$$.

What are eigenvalues and eigenstate?

 

[NFuller]

Are you familiar with linear algebra? A Vector $\mathbf{x}$ which has the property that when multiplied by a matrix $\mathbf{A}$ as
$$\mathbf{A}\mathbf{x}=\alpha\mathbf{x}$$
produces the same vector $\mathbf{x}$ multiplied by a scalar $\alpha$ is called an eigenvector. The scalar $\alpha$ is called an eigenvalue. This is analogous to the time independent Schrödinger equation
$$\hat{H}\phi(\mathbf{x})_{n}=E_{n}\phi(\mathbf{x})_{n}$$
where $E_{n}$ is an eigenvalue and $\phi(\mathbf{x})_{n}$ is an eigenstate or eigenfunction.

 

[Wrichik Basu]

Are you familiar with linear algebra? A Vector $\mathbf{x}$ which has the property that when multiplied by a matrix $\mathbf{A}$ as
$$\mathbf{A}\mathbf{x}=\alpha\mathbf{x}$$
produces the same vector $\mathbf{x}$ multiplied by a scalar $\alpha$ is called an eigenvector. The scalar $\alpha$ is called an eigenvalue. This is analogous to the time independent Schrödinger equation
$$\hat{H}\phi(\mathbf{x})_{n}=E_{n}\phi(\mathbf{x})_{n}$$
where $E_{n}$ is an eigenvalue and $\phi(\mathbf{x})_{n}$ is an eigenstate or eigenfunction.

Understood about eigenvalues. But what does an operator physically signify? What is the physical significance of energy operators?

 

[vanhees71]

Have you studied classical mechanics in the Hamilton formalism, including Poisson brackets and symmetries (Noether)? If not, chances are pretty bad that you understand what's behind the formalities of QT. So it's wise, to study classical physics to some level of sophistication first. This holds the more for classical electrodynamics and relativistic quantum field theory. Also here it's important to study the classical theory first, including a firm understanding of the relativistically covariant formulation and some representation theory of the Lorentz and Poincare group.

 

[Wrichik Basu]

Have you studied classical mechanics in the Hamilton formalism, including Poisson brackets and symmetries (Noether)? If not, chances are pretty bad that you understand what's behind the formalities of QT. So it's wise, to study classical physics to some level of sophistication first. This holds the more for classical electrodynamics and relativistic quantum field theory. Also here it's important to study the classical theory first, including a firm understanding of the relativistically covariant formulation and some representation theory of the Lorentz and Poincare group.

It's true that I didn't study classical formalism, because I'm somehow more interested in quantum. Can you recommend a book, or website, that can give me a overview of these and form a concept with which I can understand at least preliminary quantum mechanics?

 

[NFuller]

I used Classical Dynamics of Particles and Systems by Stephen T. Thornton & Jerry B. Marion. I think it's pretty good for an introductory mechanics text.

 

[Stephen Tashi]

There are some old threads about the question:
https://www.physicsforums.com/threads/physical-interpretation-of-operators-in-qm.416442/

 

[Vanadium 50]

There are lots of people who come here saying "I'm not interested in studying the middle. Take me straight to the end!" They tend not to be successful.

 

[lawlieto]

There are lots of people who come here saying "I'm not interested in studying the middle. Take me straight to the end!" They tend not to be successful.

I agree. I was really interested in quantum, and knew a lot of maths, but I simply couldn't understand it properly because I had no knowledge in classical physics. So I decided to study physics at university, I've completed my first year (lot of mechanics, EM, more maths) and even though I haven't started quantum properly yet, I'm more confident now that I'll be better at it.

 

[Wrichik Basu]

There are lots of people who come here saying "I'm not interested in studying the middle. Take me straight to the end!" They tend not to be successful.

To whom do you intend to address this to?

 

[lawlieto]

To whom do you intend to address this to?

Clearly to you, you haven't done classical mechanics and based on your posts you haven't done much linear algebra either. Believe me I'm not interested in classical mechanics either, but to do quantum properly you have to start from the beginning, and go through the steps. Of course there will be people saying classical mechanics is not necessary, and I suppose you can learn quantum without it, but having completed 1 year of undergraduate physics I'm so grateful to myself that I started from the beginning.

 

[Vanadium 50]

To whom do you intend to address this to?

To anyone who wants to go straight to one part of physics without covering the intervening material. For example, someone who wants to understand QM and only QM without any classical physics.

 

[Wrichik Basu]

To anyone who wants to go straight to one part of physics without covering the intervening material. For example, someone who wants to understand QM and only QM without any classical physics.

I was wrong to ask this question.

 

[Vanadium 50]

The question isn't wrong. The plan is wrong. You can't get to the end without going through the middle.

 

[vanhees71]

Yes, and the answers are very valuable, although maybe the OP doesn't realize it :-(.

 

[Wrichik Basu]

Anyways, I have found a book in which one can start even without knowing classical formalism. It acts as a training book and bridges the gap for one who does not know classical. I spoke with the author, described my situation, and he said that I should read that book before starting out with Griffiths. I've switched over to that book for the time being.

 

[Smalde]

May I ask what the name of said book is? It is true that one can do quantum mechanics to some extent without having seen either the Lagrangian nor the Hamiltonian formulation of quantum mechanics: engineers doing introductory courses on nanotechnology do it, but it does require some acceptance, i.e. accepting that one will not be able to truly understand the origin of many formulae. Having said this I do think that one should pursue the study of the things that one finds interesting. Maybe on the way you will interest yourself in classical mechanics and will try to understand them as well in order to more fully understand what you are reading.
So I encourage you to read as many physics as you can, but know that the way will be long.

Also knowledge of linear algebra and analysis is recommended.

 

[ftr]

I'm a beginner in QFT, starting out with D. J. Griffiths' book in this topic.

I have a question on the operators used in QM. What are operators? What is the physical significance of operators? I can understand that $\frac {d}{dt}$ to be an operator, but how can there be a total energy operator, a potential energy operator, and what is the meaning of subtracting them to give a Hamiltonian? ($\hat H = \hat T - \hat V$)

IMHO always check Google first so you can get some info before you formulate your question.

https://en.wikipedia.org/wiki/Operator_(physics)

 

[bob012345]

I'm a beginner in QFT, starting out with D. J. Griffiths' book in this topic.

I have a question on the operators used in QM. What are operators? What is the physical significance of operators? I can understand that $\frac {d}{dt}$ to be an operator, but how can there be a total energy operator, a potential energy operator, and what is the meaning of subtracting them to give a Hamiltonian? ($\hat H = \hat T - \hat V$)

Operators simply represent performing mathematical operations on the wave function which give physical aspects of the system. If you are interested in the formal structure of QM then by all means study classical mechanics first. If you are just interested in doing QM then a book like J.J. Sakurai's Modern Quantum Mechanics is good because he doesn't bother with historical development of QM but rather, just what it is and how to use it.

 

[noir1993]

I'm a beginner in QFT, starting out with D. J. Griffiths' book in this topic.

Griffiths has written books on introductory quantum mechanics, particle physics, and classical electrodynamics. I am unaware of this book on quantum field theory.

 

[bhobba]

Griffiths has written books on introductory quantum mechanics, particle physics, and classical electrodynamics. I am unaware of this book on quantum field theory.

I am pretty sure he means QM - not QFT.

But just as a suggestion I always say start with Susskind - both books:
https://www.amazon.com/dp/0465075681/?tag=pfamazon01-20
https://www.amazon.com/dp/0465062903/?tag=pfamazon01-20

The first, about classical physics is a prerequisite for the second.

Then I think you can tackle Landau for more advanced classical mechanics:
https://www.amazon.com/dp/0750628960/?tag=pfamazon01-20

Some say its not an easy read. I actually disagree with that. Its like fine wine - it's not hard to follow if you have a smattering of knowledge of multivariate calculus eg
https://www.amazon.com/dp/0471827223/?tag=pfamazon01-20

But with each reading you gain deeper and deeper insights. And right from the start you learn what an inertial frame really is - I never found the usual definitions eg a frame of reference in which a body remains at rest or moves with constant linear velocity unless acted upon by forces: any frame of reference that moves with constant velocity relative to an inertial system is itself an inertial system - at all satisfactory. With Landau its all clear but I won't spoil the enjoyment of this revelation by spelling it out here - besides you will understand it better. Of course if anyone starts a thread I will answer directly - but please do read the book.

After Laudau you are well and truly prepared for QM - after Susskind I like Sakurai - Modern QM which is now available at a good price for the paperback version:
https://www.amazon.com/dp/9332519005/?tag=pfamazon01-20

I used to suggest QM Demystified but it really has a lot of problems eg its axioms of QM are basically a crock and IMHO confuse rather than illuminate. I had a long discussion with the author about that - let's just say we agreed to disagree.

Beyond that of course is Ballentine whose 2 axioms are both succinct and spot on - but not before Sakurai.

Thanks
Bill

 

[bhobba]

I have a question on the operators used in QM. What are operators? What is the physical significance of operators?

In time grasshopper - in time.

For now just accept what Griffiths says.

Lest you think it's a cop out here is the full technical answer - but probably gobbledygook at your level (see post 137):
https://www.physicsforums.com/threads/the-born-rule-in-many-worlds.763139/page-7

Just look at it for now, don't worry if you don't understand it - its simply to reassure yourself there is an answer.

You will be able to follow it after Ballentine.

Thanks
Bill

 

[entropy1]

@bhobba does Griffith come in somewhere?

 

[bhobba]

@bhobba does Griffith come in somewhere?

Griffiths is very good, but I like Sakurai a little better.

You most certainly could study it between Susskind and Sakurai and nowadays is quite cheap as well:
https://www.amazon.com/dp/1316646513/?tag=pfamazon01-20

Much cheaper than when I got my copy all those years ago along with his also excelllent book on EM.

It really is good that some of the higher quality books on QM are now quite cheap.

Ballentine is more expensive, but IMHO still the best - although not the first book you should study:
https://www.amazon.com/dp/9810241054/?tag=pfamazon01-20

Thanks
Bill