Hamiltonian: The Key to Understanding Total Energy in Quantum Mechanics

[Barny]

Hello!

I've just finished a discussion with my peers and lecturer on some of the postulates of QM,

My lecturer said that "the total energy of any quantum mechanical system is always given by the Hamiltonian, which may be obtained by appropraite application of the classical Hamiltonian"

I took issue with this, there are examples within classical mechanics where the Hamiltonian is conserved but the total energy of a system isn't. I therefore fail to see how the Hamiltonian can always represent the total energy of QM system.

Am I completely off the mark here?

Kind regards
Barny

 

[janakiraman]

well i had thought that the hamiltonian always depicted the energy of the system and one of the fundamental postulates of the classical mechanics is conservation of energy. I don't understand what u mean by hamiltonian is conserved but not the energy

 

[Barny]

Hi, thanks for replying.

It's possible to construct systems in classical mechanics where the Hamiltonian is conserved but the total energy of the system is not. In this instance the Hamiltonian no longer represents the total energy of the system.

For example on Section 6.4, page 18 of http://www.hep.phys.soton.ac.uk/~trmorris/teaching/undergrad/PHYS3007.html discusses this.

My argument is that if you use the same construct of the Hamiltonian in quantum mechanics as you do classical mechanics i.e

$$H=\sum$$$$\frac{\partial L}{\partial q^{.}}q^{.}-L$$

Then surely the same constraints apply to it, as in the example posted above. Does this not mean that there may be an instance within QM where the Hamiltonian is not the true representation of the total energy? Or have I missed the point?

Kind regards
Barny

 

[olgranpappy]

Hello!

I've just finished a discussion with my peers and lecturer on some of the postulates of QM,

My lecturer said that "the total energy of any quantum mechanical system is always given by the Hamiltonian, which may be obtained by appropraite application of the classical Hamiltonian"

Another interesting issue is the existence of terms in the quantum hamiltonian that vanish in the classical limit. For example, spin. Obviously these can not be obtained in any reasonable way from just the classical hamiltonian.

I took issue with this, there are examples within classical mechanics where the Hamiltonian is conserved but the total energy of a system isn't. I therefore fail to see how the Hamiltonian can always represent the total energy of QM system.

Am I completely off the mark here?

Kind regards
Barny

 

[jomoonrain]

Hello!

I've just finished a discussion with my peers and lecturer on some of the postulates of QM,

My lecturer said that "the total energy of any quantum mechanical system is always given by the Hamiltonian, which may be obtained by appropraite application of the classical Hamiltonian"

I took issue with this, there are examples within classical mechanics where the Hamiltonian is conserved but the total energy of a system isn't. I therefore fail to see how the Hamiltonian can always represent the total energy of QM system.

Am I completely off the mark here?

Kind regards
Barny

well, i think they are both right.
but the first statement, in my opinion, should better be related as follows: if we want to describe a physical system, we HAVE TO find its hamiltonian,which represents the total energy of the system, besides this method, no others exits.
in classical mechanics,hamiltonian do not allways represents the total energy, but it is the case that the system are not isolated or closed.In another words,the system we observed is in fact part of a more biger one. so in classical mechanics,we can allways (in principle)find the hamiltonian(representing the total energy) of a system,or find a system whose hamiltonian represents it total energy.
hope this will be helpful to you.