HUP and good quantum numbers that commute

In summary, the Heisenberg Uncertainty Principle (HUP) only applies to conjugate attributes that do not commute, such as position and momentum. However, for good quantum numbers that do commute, the HUP does not apply to simultaneous measurements. This means that for the hydrogen atom, the total energy and angular momentum can be known simultaneously, but not individual angular momenta. The set of good quantum numbers uniquely designate a particular quantum state, so they must be known simultaneously. As for the double slit experiment, it is possible to know the combined set of good quantum numbers at both slits simultaneously, as long as the quantum state is known when it enters the 3D volume of each slit.
  • #1
Salman2
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I have a question about the HUP. As I understand the HUP, it only applies to conjugate attributes that do not commute, such as position and momentum. However, many good quantum numbers do commute, so does this mean that the HUP does not apply to simultaneous measurement of such good quantum numbers ?

Also, for the hydrogen atom, is it not true that the total energy Hamiltonian, and angular momentum commute, thus the HUP would not apply to their simultaneous measurement ?
 
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  • #2
The set of good quantum numbers uniquely designate a particular quantum state. Therefore, you have to be able to know them simultaneously. By the way, quantum numbers themselves do not "commute", it is the operators relate to them that do (or do not...).

So yes, you can simulatneously know the total energy of the hydrogen atom and its total angular momentum. But that does not mean that you can also know individual angular momenta, such as electronic orbital angular momentum. That will depend on what terms you actually consider in the Hamiltonian (spin-orbit coupling, hyperfine interaction, ...)
 
  • #3
DrClaude said:
The set of good quantum numbers uniquely designate a particular quantum state. Therefore, you have to be able to know them simultaneously. ..So yes, you can simultaneously know the total energy of the hydrogen atom and its total angular momentum.
Thanks.

I would like to follow-up your reply to a question about the double slit experiment. At the moment in time a quantum state enters the 3D volume of each slit, would it be correct to say that we can 'know' simultaneously all good quantum numbers for the state at each 3D slit space ? If yes, can we then know the combined set of good quantum numbers at both slits, simultaneously ?
 

1. What is the Heisenberg Uncertainty Principle (HUP)?

The Heisenberg Uncertainty Principle is a fundamental principle in quantum mechanics that states that it is impossible to simultaneously know the exact position and momentum of a particle. This means that the more precisely we know the position of a particle, the less precisely we can know its momentum, and vice versa.

2. How does the HUP relate to good quantum numbers?

The HUP is closely related to good quantum numbers because it restricts the precision with which we can measure certain properties of a particle. Good quantum numbers are measurable properties of a quantum system that do not change over time, and therefore commute with the Hamiltonian operator. This means that good quantum numbers can be precisely measured without violating the HUP.

3. Why is it important for good quantum numbers to commute?

Good quantum numbers need to commute with the Hamiltonian operator because this ensures that they remain constant over time. If a quantum number does not commute with the Hamiltonian, it will change over time and will not be a good quantum number. This can lead to incorrect predictions and interpretations of quantum systems.

4. What are some examples of good quantum numbers?

Examples of good quantum numbers include spin, angular momentum, and energy. These properties are conserved in a system and therefore commute with the Hamiltonian operator. Other examples may include quantum numbers related to symmetries of a system, such as parity or charge.

5. How does the HUP impact our understanding of the quantum world?

The Heisenberg Uncertainty Principle challenges our traditional understanding of the world, as it shows that there are inherent limitations on our ability to measure and predict the behavior of particles at the quantum level. It highlights the probabilistic nature of quantum mechanics and forces us to rethink our classical notions of cause and effect. The HUP is a fundamental principle that has shaped our understanding of the quantum world and continues to play a crucial role in modern physics.

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