# Why isn't the constant force problem in QM classes?

pellman
I have not been able to find the quantum constant force problem in the various QM texts that I have checked. I am certain it was not covered in any of the classes I took. Why is that? I would think it would be next thing covered after free particles.

## Answers and Replies

Homework Helper
There's no such thing as "force" in QM.

Daniel.

P.S. A linearly coordinate depending potential energy operator (in the Schrödinger picture, Dirac fomulation) (let's say coming from a constant and homogenous gravitational field of a massive pointlike particle) has a discrete spectrum of eigenvalues and the Airy functions as eigenfunctions.

QMrocks
You mean solving of Schrodinger equation with a constant force?

Homework Helper
There's no force in the SE, get over it.

Daniel.

QMrocks
Physically, u just need to have a position dependent potential.

Staff Emeritus
Gold Member
Have you never come across a triangular well ? I think that's what you are looking for.

Homework Helper
QMrocks said:
Physically, u just need to have a position dependent potential.

Not necessarily. Think of a free particle acted upon by a time dependent perturbation to the Hamiltonian.

Daniel.

Homework Helper
Gold Member
There's no force in the SE, get over it.

But $F = -\nabla U$

and I believe U appears in the Schrodinger equation.

This is true; however as you know the non-relativistic Schrodinger equation is a scalar-valued equation. To be analogous to Newton's laws directly, you would have to have a vector-valued function in a differential equation that becomes the Schrodinger equation with some mathematical trickery. If you want to follow the classical prescription.

Staff Emeritus
Gold Member
pellman said:
I have not been able to find the quantum constant force problem in the various QM texts that I have checked. I am certain it was not covered in any of the classes I took. Why is that? I would think it would be next thing covered after free particles.

It is the triangular potential, and it is used, because the Earth gravity field (near its surface) does this. I have a colleague here at the ILL which plays with bouncing cold neutrons in the gravity field and he studies their quantum properties, V. Nesvizhevsky. Do a search on his name, you'll find some papers about it.

cheers,
Patrick.

QMrocks
My lecturer worked out the problem of Schrodinger equation with constant force term (i.e FX, where F is a the constant value of the force and X is the postition operator) in his notes, just by using the Heisenberg equation of motion.

Also in electron transport problems in devices, one have to deal with electron transport with a triagular potential, but with open boundary.

QMrocks
dextercioby said:
There's no such thing as "force" in QM.

Daniel.

P.S. A linearly coordinate depending potential energy operator (in the Schrödinger picture, Dirac fomulation) (let's say coming from a constant and homogenous gravitational field of a massive pointlike particle) has a discrete spectrum of eigenvalues and the Airy functions as eigenfunctions.

Sorry, i do not understand this concept. Does using the concept of 'force' (as in the gradient of potential V) in QM leads to serious contradictions in theory? Maybe when one try to deal with vector potential A in EM?

Homework Helper
There is no force, there are only observables which are mathematical objects (1). "Force" is the key concept in the Newtonian formulation of classical mechanics. However, to apply the quantization scheme proposed by Dirac (2), on has to use the Hamiltonian formulation of classical dynamics. J.Schwinger (3) proposed a construction which starts from the Lagangian action. However, classically, before quantization, such concepts as "momentum, force, acceleration, velocity" are basically useless when applying the quantization postulate.

Daniel

Notes
(1) densly defined selfadjoint linear operators acting on the (rigged, if unbounded) separable Hilbert space of states.
(2) Graded Dirac bracket on the reduced phase space goes to $$\frac{1}{i\hbar}$$ times graded Lie gracket.
(3) See R. Newton "Quantum Physics for Graduate Students", Springer Verlag.

Observable
convert it into potential

Constant force can be treated as linear potential. Then you can solve it sectionally. A similar problem has been investigated: V(x)=ABS(x). The solution to this potential is just Airy functions. But it doesn't have as much practical value as the ones show up in QM books(like oscillator potential and coulomb potential) and usully linear effect could be easily dealt with perturbation theory.You can check it in some mathematical handbook for Airy functions.

Staff Emeritus
Observable said:
Constant force can be treated as linear potential. Then you can solve it sectionally. A similar problem has been investigated: V(x)=ABS(x). The solution to this potential is just Airy functions. But it doesn't have as much practical value as the ones show up in QM books(like oscillator potential and coulomb potential) and usully linear effect could be easily dealt with perturbation theory.You can check it in some mathematical handbook for Airy functions.

Surprisingly enough, the linear potential IS quite widely used, especially in many tunneling barrier model. One of the most common application is in the Fowler-Nordheim model of field emission (see, for example, http://ece-www.colorado.edu/~bart/book/msfield.htm). The simplest approximation to such a phenomenon is a triangular barrier. This is the model most people use to describe field emission, and if you know what that is, you'll know that your plasma screen and many other devices make use of this effect.

Zz.

Observable
ZapperZ said:
Surprisingly enough, the linear potential IS quite widely used, especially in many tunneling barrier model. One of the most common application is in the Fowler-Nordheim model of field emission (see, for example, http://ece-www.colorado.edu/~bart/book/msfield.htm). The simplest approximation to such a phenomenon is a triangular barrier. This is the model most people use to describe field emission, and if you know what that is, you'll know that your plasma screen and many other devices make use of this effect.

Zz.
I think those applications are actually treated by WKB approximation, so comes the linear potential

Staff Emeritus
Observable said:
I think those applications are actually treated by WKB approximation, so comes the linear potential

Yes, but it is STILL a linear potential barrier.

The WKB approximation simply allows one to already "pre-determine" the wavefunction, rather than solving for one explicitly. In many cases, this is a detail that isn't necessary since all one cares about is the transmission probability that determine the current density. Thus, the WKB wavefunction is good enough.

Zz.

Staff Emeritus
Gold Member
Besides FETs, there's a whole host of other places where you see approximations to triangular wells or linear potentials - in modulation doped heterostructures, metal-semiconductor interfaces, biased junctions, etc.

And in most of these places, the linearity of the potential does not come out of the approximation used to model the system (in any case, such an approximation is chosen for a reason) but is the expected/designed profile.

aav
pellman said:
I have not been able to find the quantum constant force problem in the various QM texts that I have checked.

See J. J. Sakurai, Modern Quantum Mechanics, revised ed. Problem 24, Chapter 2.

Staff Emeritus
Gold Member
aav said:
See J. J. Sakurai, Modern Quantum Mechanics, revised ed. Problem 24, Chapter 2.
And the bit about gravity-induced interference between neutron beams (also in Ch. 2)

Observable said:
I think those applications are actually treated by WKB approximation, so comes the linear potential

I should point out here that although almost everyone does use the WKB approximation in these cases, there has been a bit of a move in recent times to be more cautious in the application of the FN equation to arrays of emitters. Look at K Jensen's papers on FE - see especially his chapter in Vacuum Microelectronics.

Staff Emeritus
Kane O'Donnell said:
I should point out here that although almost everyone does use the WKB approximation in these cases, there has been a bit of a move in recent times to be more cautious in the application of the FN equation to arrays of emitters. Look at K Jensen's papers on FE - see especially his chapter in Vacuum Microelectronics.

Yes, I'm aware of that, since I've talked to him a few times and have been looking into why people are using the FN model BEYOND the region of applicability, such as in a high-gradient photoinjector.

I think Kevin and I share a common "bond" - we're both refugees from condensed matter working in accelerator physics. :)

Zz.

DaTario
Regardig what most of you have said:

is acceleration and mass observables? If so, why force is not an observable ?
Even in Heisenberg picture, equations with operators like force never show up?