Math connections between QM and classical?

[Farn]

If you were faced with an easy kinematics problem (trajectory of a baseball) could you use QM instead of classical mec. to get a result that’s just as accurate and useful?

I don’t know how well the maths of QM simplify, but if you were to set up the above problem in QM would you end up being able to essentially simplify the math down to the well known classical equations that describe the balls motion?

The only point of these questions is to help me to realize the connections between these two seemingly very different ways of looking at things (QM and classical). I feel there must be some connection because at one scale they should both give the same exact results.

 

[Tyger]

The Lagrangian function is the most general connection between classical and quantum physics. It involves the action integral instead of the classical force concept, which is a derivative of the energy. Derivative is meant in the calculus sense of the word.

A good description is the "Special Lecture" in Feynman's Lectures.

 

[Integral]

Tajectory is not a quantum concept, no you cannot compute the trajectory of a baseball in QM.


You could compute the probability that the ball would tunnel through the bat, but not where the ball would land if the ball did not tunnel.

 

[Farn]

Tajectory is not a quantum concept, no you cannot compute the trajectory of a baseball in QM.

Hmm, I guess I was mislead at some point. I was under the idea that QM was a theory that came about to fix the problems that classical had at the atomic level, but that it could also be used in place of all classical equations... sort of like an all-in-one tool. If this is not the case, than QMs usefulness is limited to the particle world, just as classical is limited to more massive objects. This a correct assumption?

Anyway, so your saying there is no way to get trajectory or the like out of QM?

 

[Integral]

That is correct, QM is statistical in nature it simply does not deal with trajectrories. It deals with atomic interactions, that is the quantum world after all.

 

[Tyger]

You can use the classical Lagrangian to solve the baseball problem, and it has the most direct connection to Quantum Mechanics. Solving the baseball problem with QM would be like using an A-bomb to kill a gnat, but you could do it in principle.

 

[Farn]

You can use the classical Lagrangian to solve the baseball problem, and it has the most direct connection to Quantum Mechanics.

Not to make too big a deal out of this, but Tyger, does this mean you disagree with Integrals reply?

Tajectory is not a quantum concept, no you cannot compute the trajectory of a baseball in QM.

 

[jcsd]

You couldn't find the trajectory of baseball as it involves gravity, something which quantum field theory explains poorly. You could consider the wave function of a baseball, but you would expect ot to display classical behaviour and considering the baseball as a collection of particles would lead to decoherence.

 

[Hurkyl]

QM can handle ordinary potentials, though, can't it? If so then one can impose the typical U=mgh potential to approximate the gravitational field on the Earth's surface.

 

[Integral]

While the Lagrangian is used in QM it is also a very handy tool to handle classical problems. Use of the Lagrangain does not inherently equal QM. It is simply a way of solving dynamic problems using energy considerations.

I will stick by my initial statement trajectrories are not a part of QM.

 

[futz]

I would tend to agree with Integral.

Classicaly, the trajectory of an object is simply the path that it follows, or more precisely the position of the object as a function of time. If we know the initial velocity of the ball and the angle it makes with the ground, we can easily calculate the trajectory -- this allows us to predict with certainty where the ball will be at any instant in time after it was hit.

If we extend the idea of the path of the ball to QM, then the ball does not travel a single, well-defined path -- instead, it travels all paths simultaneously, as described by its wavefunction. Now, some of these paths would be more preferable than others (they have higher probability, again calculated from the wavefunction). And, the most preferred path (highest probability) would correspond to our classical trajectory. However, since all paths are included, it is impossible for us to know where an object will be and how fast it will be going at any specific moment in time, contrary to the classical picture.

Ultimately, it just doesn't make sense to me to try and apply a statistical theory based on probabilities and expectation values to a macroscopic object with well-defined properties.