Can Quantum Mechanics Have 'Elegant' Trajectories in Hilbert Space?

In summary, the conversation discusses the use of state-space trajectories in classical and quantum mechanics to represent the time evolution of a given system. While classical mechanics uses ellipses to represent the motion of a harmonic oscillator, it is unclear if a similar representation can be used in quantum mechanics due to the complex nature of wave functions. The conversation also mentions the use of unit vectors and projective Hilbert space for representing trajectories in quantum mechanics. A book recommendation is given, but it is noted that the topic is considered difficult by mathematicians.
  • #1
Logic Cloud
23
0
State-space trajectories in classical mechanics can be used to nicely represent the time evolution of a given system. In the case of the harmonic oscillator, for instance, we get ellipses. How does this situation carry over to quantum mechanics? Can the time evolution of, say, the quantum harmonic oscillator be represented by some 'trajectory' in Hilbert space? The obtained wave functions do not seem to lend themselves for such 'elegant' representations.
 
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  • #2
If you choose a unit vector, say for spin 1/2, the trajectory in Hilbert space is just rotation of a unit vector. It's very elegant.
 
  • #3
There is also the trajectories in the projective hilbert space and their hamiltonian dynamics and poisson brackets.
 
  • #4
Logic Cloud said:
State-space trajectories in classical mechanics can be used to nicely represent the time evolution of a given system. In the case of the harmonic oscillator, for instance, we get ellipses. How does this situation carry over to quantum mechanics? Can the time evolution of, say, the quantum harmonic oscillator be represented by some 'trajectory' in Hilbert space? The obtained wave functions do not seem to lend themselves for such 'elegant' representations.

If that sort of thing interests you here is the book to get:
https://www.amazon.com/dp/0387493859/?tag=pfamazon01-20

Be warned - its what mathematicians call decidedly non-trivial - meaning it's hard.

Said with a German accent - We have ways and means of stopping you asking certain questions :-p:-p:-p:-p:-p:-p:-p:-p:-p

Thanks
Bill
 
Last edited by a moderator:

1. What is a Trajectory in Hilbert Space?

A trajectory in Hilbert space refers to the path or motion of a quantum system in the mathematical space known as Hilbert space. It is a fundamental concept in quantum mechanics and is used to describe the evolution of a quantum system over time.

2. How is Hilbert Space used in Trajectories?

Hilbert space is used to mathematically describe the states and evolution of quantum systems, including trajectories. It provides a framework for representing the dynamics of quantum systems and calculating the probabilities of different outcomes.

3. What is the significance of Trajectories in Hilbert Space?

The concept of trajectories in Hilbert space is significant because it allows scientists to study and understand the behavior of quantum systems, which can be highly complex and unpredictable. It also plays a crucial role in the development of quantum technologies and applications.

4. How are Trajectories in Hilbert Space different from Classical Trajectories?

Trajectories in Hilbert space are fundamentally different from classical trajectories because they describe the behavior of quantum systems, which behave differently from classical systems. In Hilbert space, a quantum system can exist in multiple states simultaneously, and its trajectory may not follow a well-defined path.

5. What are some real-world applications of Trajectories in Hilbert Space?

Trajectories in Hilbert space have various applications, including quantum computing, quantum cryptography, and quantum sensing. They also play a role in understanding and predicting the behavior of quantum systems in fields such as materials science, chemistry, and biology.

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