Transition: Classical Mechanics to Quantum Mechanics

In summary, the process of quantizing a classical system involves setting the commutator [q_a, p_a] = i, replacing the Poisson bracket {q_a, p_a} = 1. This step is crucial and serves as the cornerstone of the quantization process. One motivation for this step is that it works for simple one-particle systems in Cartesian coordinates. However, this replacement is not the end of the story, as a consistent ordering of the operators must also be decided.
  • #1
go quantum!
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0
Imagine that I have a system that is described classically by a given Hamiltonian which is a function of a given set of parameters [tex]q[/tex] and their canonical conjugate momenta [tex]p=\frac{\partial L}{\partial \dot{q}}[/tex].
Then, I will say that the quantum description of the same system is guided by setting the commutator [ tex ] [q_a,p_a]=i [ /tex ] because the Poisson bracket is [tex ]{q_a,p_a}=1[ /tex ].

This step is crucial and it is the cornerstone of the process of quantizing. I would like to ask if you know some motivations for this step. Do you understand it?

Thanks for you help!
 
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  • #2
The motivation for replacing Poisson brackets by commutators is that it works for simple one-particle systems in Cartesian coordinates - {x, p} → [x, p].

Note, that this replacement is not the end of the story. Now that the Hamiltonian is given in terms of operators, you have to decide a consistent ordering for those operators.
 
  • #3
Is that the only motivation? It can't be...
 

1. What is the difference between classical mechanics and quantum mechanics?

Classical mechanics is a physical theory that describes the motion of macroscopic objects such as planets, cars, and baseballs. It is based on Newton's laws of motion and the concept of determinism, where the future state of a system can be predicted from its initial conditions. On the other hand, quantum mechanics is a theory that describes the behavior of particles at the microscopic level, such as atoms and subatomic particles. It is based on the principles of uncertainty and probability, where the behavior of particles is not predictable with certainty.

2. Why is it important to transition from classical mechanics to quantum mechanics?

The transition from classical mechanics to quantum mechanics is important because it allows us to understand the behavior of particles at the microscopic level, which is crucial in fields such as quantum physics, chemistry, and materials science. It also provides a more accurate description of nature, as classical mechanics fails to explain certain phenomena, such as the behavior of subatomic particles.

3. What are some key concepts in quantum mechanics?

Some key concepts in quantum mechanics include wave-particle duality, superposition, uncertainty principle, and entanglement. Wave-particle duality states that particles can exhibit both wave-like and particle-like behavior. Superposition is the principle that a quantum system can exist in multiple states simultaneously. The uncertainty principle states that it is impossible to know the exact position and momentum of a particle at the same time. Entanglement is a phenomenon where two or more particles become connected and share properties, even when separated by large distances.

4. How is the mathematical framework of quantum mechanics different from classical mechanics?

The mathematical framework of quantum mechanics is different from classical mechanics in that it uses complex numbers and operators to describe the behavior of particles. Classical mechanics, on the other hand, uses real numbers and equations to describe the motion of macroscopic objects. Additionally, in quantum mechanics, the state of a system is described by a wave function, which contains all the information about the system, whereas in classical mechanics, the state of a system is described by its position and momentum.

5. What are some real-world applications of quantum mechanics?

Quantum mechanics has many real-world applications, including transistors, lasers, and cryptography. Transistors, which are used in electronic devices, are based on the principles of quantum mechanics. Lasers, which are used in various industries, use quantum mechanics to amplify light and produce a coherent beam. Cryptography, which is used to secure information and communication, relies on the principles of quantum mechanics, such as entanglement, to create unbreakable codes.

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