Math connections between QM and classical?

In summary, QM cannot be used to accurately and usefully compute the trajectory of a baseball like classical mechanics can, as it deals with statistical probabilities rather than specific paths. Additionally, while the Lagrangian function is a connection between the two theories, it does not necessarily mean that QM can be used in place of all classical equations.
  • #1
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.
 
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  • #2
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.
 
  • #3
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.
 
  • #4
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?
 
  • #5
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.
 
  • #6
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.
 
  • #7
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.
 
  • #8
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.
 
  • #9
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.
 
  • #10
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.
 
  • #11
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.
 

1. How are quantum mechanics and classical mechanics related?

Quantum mechanics and classical mechanics are two different theories used to describe the behavior of particles. Quantum mechanics is used to describe the behavior of particles at a microscopic level, while classical mechanics is used to describe the behavior of particles at a macroscopic level. However, at the macroscopic level, quantum mechanics reduces to classical mechanics, meaning that classical mechanics is a special case of quantum mechanics.

2. What are some key mathematical concepts that connect quantum mechanics and classical mechanics?

One key mathematical concept that connects quantum mechanics and classical mechanics is the principle of superposition. This states that a quantum system can exist in multiple states simultaneously, while a classical system can only exist in one state at a time. Additionally, the mathematical concept of wave-particle duality connects the two theories, as particles in quantum mechanics can exhibit both wave-like and particle-like properties.

3. How does the Heisenberg uncertainty principle relate to classical mechanics?

The Heisenberg uncertainty principle states that it is impossible to know both the position and momentum of a particle with absolute certainty. In classical mechanics, the position and momentum of a particle can be known with absolute certainty at any given time. This shows that the Heisenberg uncertainty principle is a unique feature of quantum mechanics and does not apply to classical mechanics.

4. What role do probability distributions play in connecting quantum mechanics and classical mechanics?

In quantum mechanics, the behavior of particles is described by wave functions, which are probability distributions that describe the likelihood of finding a particle in a certain state. In classical mechanics, probability distributions are also used to describe the behavior of particles, such as in the kinetic theory of gases. This shows that probability distributions play a key role in connecting the two theories.

5. How does the concept of measurement in quantum mechanics relate to classical mechanics?

In quantum mechanics, the act of measurement can change the state of a particle, and the outcome of a measurement is probabilistic. In classical mechanics, the act of measurement does not affect the state of a particle, and the outcome of a measurement is deterministic. This shows that the concept of measurement in quantum mechanics is fundamentally different from classical mechanics.

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