Is Probability in Quantum States Proportional to Energy Levels?

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SUMMARY

The discussion centers on the relationship between quantum states and their probabilities, specifically questioning whether the probability of a particle being in a certain state is proportional to its energy level. It is established that probabilities in quantum mechanics (QM) do not reflect the likelihood of being in a specific state but rather the probabilities of obtaining measurement results from a prepared state. The concept of eigenstates is emphasized, with the conclusion that if a system is prepared in a definite energy eigenstate, the probability of measuring any other energy is zero.

PREREQUISITES
  • Understanding of quantum mechanics principles, particularly eigenstates and measurement.
  • Familiarity with the Schrödinger equation and energy quantization in quantum systems.
  • Knowledge of statistical mechanics, including the Boltzmann distribution.
  • Basic grasp of quantum state preparation and its implications on measurement outcomes.
NEXT STEPS
  • Study the implications of eigenstates in quantum mechanics and their role in measurement outcomes.
  • Learn about the Boltzmann distribution and its application in statistical mechanics for systems in thermal equilibrium.
  • Explore the concept of mixed states and their differences from pure states in quantum mechanics.
  • Investigate the effects of external interactions on quantum systems and how they influence energy measurements.
USEFUL FOR

Students and professionals in physics, particularly those focusing on quantum mechanics, quantum state preparation, and statistical mechanics. This discussion is beneficial for anyone looking to deepen their understanding of measurement theory in quantum systems.

  • #31
bob012345 said:
I understand we can only measure eigenvalues. But we can certainly know the expectation value for the energy of the system.

If what you mean is "expectation value", then you need to say "expectation value". You can't just say "energy" and expect everyone to know that you mean "the expectation value of energy in some particular state that is not an eigenstate of energy".

bob012345 said:
I'm just trying to understand what that entails

You have been told two different ways what kind of preparation is necessary for a Boltzmann distribution to apply. I did in post #15, and @f95toli did in post #17. The two descriptions we gave are equivalent.

bob012345 said:
and how it would apply to a quantum system of a particle in a box?

It wouldn't. You have already been told why (@Nugatory did that in post #14).
 
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  • #32
The original question of this thread has been answered and I think I have synthesized in my mind all the answers given about that and the discussion of the Boltzmann distribution so I thank everyone for their answers.
 
  • #33
bob012345 said:
The original question of this thread has been answered and I think I have synthesized in my mind all the answers given about that and the discussion of the Boltzmann distribution so I thank everyone for their answers.

Thanks for the feedback! I'm glad we were able to answer your questions.
 
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