Use of potentials in Quantum theory

[spaghetti3451]

In quantum mechanics, we rarely talk in terms of forces and acceleration. Instead, we work with potentials and energy, and we restrict ourselves to conservative systems where the energy is constant.

Why?

There are several alternative formulations of classical mechanics, all of which are based on the same physical principles and all of which give the same answer. One in particular is called the Hamiltonian method.

Why have several alternative forumlations? What are the (same) physical principles?
What does the Hamiltonian method do for us?

 

[spaghetti3451]

In QM, the equivalent of the Hamiltonian, called the Hamiltonian operator, plays a fundamental role.

Why does the Hamiltonian operator play a fundmanetal role in QM? I mean, why has QM been constructed using operators and using analogies from Hamiltonian Mechanics?

In principle, the Hamiltonian method is just a different mathematical technique but of course
we know the momentum p does have physical significance too so it can give different physics insights.

I don't understand.

The two methods are equivalent and the choice is really convenience for any given
system. The Hamiltonian method can be used for other coordinate systems quite easily and
even other pairs of variables than x and p.

Please explain for God's sake.

Most importantly for our purposes, when comparing with quantum mechanics, the Hamiltonian approach gives equations much closer to those in quantum mechanics, hence it is good to know of its existence.

Gives eqns much closer to those in QM?!

 

[dextercioby]

Let's see what's peculiar of classical mechanics in the Newtonian formulation that we would normally not expect to see in a theory meant to describe microscopic interactions ?

Answer this question and for this I'll post the Newton's equations in differential form for a pointlike object

$$m\ddot{\vec{r}}(t) = \Sum \vec{F}(\vec{r},\dot{\vec{r}},t)$$

They should hint you towards the answer I've asked you for in first paragraph.

 

[spaghetti3451]

I still don't understand!

 

[dextercioby]

So I've asked you to say a feature of Newtonian mechanics that would not be expected in microscopic/quantum physics and through elimination of which, the normal Hamiltonian formalisms of both classical mechanics and quantum mechanics can be exhibited.

 

[spaghetti3451]

A feature of Newtonian mechanics that would not be expected in microscopic/quantum physics:

That we cannot predict the position of a particle with absolute certainty unless we decide to make the momentum infinitely uncertain?
I guess that this implies that the momentum and hence the force on the particle can't be calculated. This argument is hand-waving, so I beg you to give me the answer.

 

[dextercioby]

No, I meant the presence of velocity-dependent forces/interactions, for example a particle in free-fall facing air-friction whose slowing force can be aproximated to be proportional to the speed the particle is falling (or to its square, if the fluid is more viscous).

In these cases, the Newton's equations can easily be shown to be equivalent to Hamilton's equations. If the interaction forces are also time-independent, then the Newton's equations become simpler and the system posesses motion integrals like momentum, angular momentum and especially energy.

In this case, the Hamilonian is simply the energy of the particle as a sum between the kinetic energy and the potential energy defined as the potential deriving from the resulting force as a line (curvilinear) integral.

About the Hamiltonian's fundamental role: well, in classical but also in the quantum dynamics, if the Hamiltonian is not explicitely time-dependent, it is the generator of symmetry transformations on the physical states, namely time translation/evolution.

In certain formulations of QM, the existence Hamiltonian operator is postulated together with the Schroedinger equation which shows according to which law the mathematical objects used to describe physical states of the systems evolve in time.

In symmetry-based axiomatization of QM, both the Hamiltonian and the SE become derived concepts from another axiomatic structure.

 

[Gokul43201]

<Off-topic> Dex, welcome back (as the green monster)! </OT>

 

[A. Neumaier]

In quantum mechanics, we rarely talk in terms of forces and acceleration. Instead, we work with potentials and energy, and we restrict ourselves to conservative systems where the energy is constant.

Why?

Only for introductory purposes. Dissipative quantum systems are much more complex, hence are not treated early on. (Not even in classical hamiltonian mechnaics - except for the damped harmonic oscillator.)

In Hamiltonian systems, force is the gradient of the potential, hence is not needed separately.