Understanding the Hamiltonian of a System: Explained for Beginners

[janakiraman]

I did search for topics in this forum. but i could not find basics that deal with hamiltonian of the system

Well I'm pretty new to the field of quantum mechanics. I just could not understand what exactly it means by the hamiltonian of a system? I was told that it describes the total energy of the system. If that is the case how does it describes the total energy? and does this total energy includes the energy for the interaction with the surroundings.

Can you please provide a small lay man example and explain the same

 

[HallsofIvy]

There are others who know more about this than I do but the Hamiltonian is just the "total energy" function, as it depends upon both position and velocity (the "state" variables).

For example, in the very simple case of a single object falling vertically under a gravity field, the total energy is just potential energy, mgx, plus the kinetic energy (1/2)m(x')²: H(x, x')= mgx+ (1/2)m(x')².

 

[janakiraman]

Hey thank you very much for your reply. But I would like to know does hamiltonian depend only on the state variables position and velocity or does it depend on other variables as well? For example if we need to take into consideration the internal energy or the electrical / magnetic energy of a system then how must we represent it in the hamiltonian?

 

[mrandersdk]

The hamiltonian in QM is a operator, which when you take the expectation value gives you the total energy. You are right that it also can include internal energies and interactions with fx. a electromagnetic field. How it looks like is totally dependent on what you are trying to describe.

It seems to me that your questions is so basic that you should read Griffiths "Introduction to Quantum Mechanics"

https://www.amazon.com/dp/0131244051/?tag=pfamazon01-20

and then maybe come back, when you have read some of that. When you understand that book you could look at the more advanced book by Sakurai, but the only way to learn it, is to get experience so the best advice is to take some classes.

 

[malawi_glenn]

also try google 'hamiltonian mechanics'

 

[janakiraman]

Thank you guys. I'm actually reading Feynman lecture's vol 3. I tried to real hamiltonian matrix chapter. But there i din understand it and so i posted. Yeah I will also try to have a look at the griffith's book. But the more i read the more the questions arise out of me :). Anyways thank you very much for the concerned reply.

 

[nicksauce]

Also note that the Hamiltonian is generally written in terms of p(momentum) and x, rather than x' and x, so one would write
H = p^2/2m + V(x). (Where V is the potential)

 

[yoyoq]

just one opinion, but i think
feynmans books are too 'showy'
and if you are really into it
griffiths is much much better.
griffiths is really good for both EM and QM.

in one sentence the hamiltonian tells you
how a system evolves in time.
this happens to be related to the energy which
is no small coincidence.

 

[humanino]

You may benefit to already strengthen your understanding of classical mechanics Before reading vol3, did you complete vol1 and vol2 ? There is one specific chapter in vol2 that you must read in any case, it's chapter 19 "the principle of least action". It's a monument in the history of physics teaching. Seriously

 

[mrandersdk]

I agree wih Yoyoq, but feynmans book is a good book, but it is best when you know the subject first, it is not so good to learn from.

 

[janakiraman]

Ok I will def give griffith a chance :). I have not read vol 1 and 2. I directly plunged into 3 due to lack of time :(. Will def read the chpt 19 of the vol 2.

Also another small basic dbt. If a system has 'n' atoms, the hamiltonian that is used to define the system will be a summation of the energies of n different atoms correct? Or is it possible to take a statistical average of the energy of the system and represent the hamiltonian using that average energy?

 

[lightarrow]

I did search for topics in this forum. but i could not find basics that deal with hamiltonian of the system

Well I'm pretty new to the field of quantum mechanics. I just could not understand what exactly it means by the hamiltonian of a system? I was told that it describes the total energy of the system. If that is the case how does it describes the total energy? and does this total energy includes the energy for the interaction with the surroundings.

Can you please provide a small lay man example and explain the same

H = p*q' - L

p = generalized momentum = (p1,p2,...pn)
q' = time derivative of canonical coordinates = (q1', q2',...qn')
* = scalar product
L = Lagrangian

 

[janakiraman]

@ lightarrow

I guess p*q' indicates the kinetic energy of the system but what i don't understand is what it means by the time derivatives of canonical coordinates?

And again i don't understand L, does that mean potential energy? If so why is there a negative sign between?

 

[nicksauce]

L is the so-called Lagragian of the system, L = T - V (T is kinetic energy, V is potential). The Hamiltonian is defined in the way lightarrow wrote, but that's not all that important... what's important is that in the case where the potential is time independent the Hamiltonian is the total energy of the system H = T + V.