- #1
entropy1
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What is the relation between "superposition" and "the Heisenberg uncertainty relation"?
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Difference/similarity between superposition and uncertainty
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- Thread starter entropy1
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Discussion Overview
The discussion explores the relationship between the concepts of superposition in quantum mechanics and the Heisenberg uncertainty principle. It addresses theoretical aspects of quantum states, measurement outcomes, and the implications of non-commuting observables.
Discussion Character
- Exploratory, Technical explanation, Conceptual clarification, Debate/contested
Main Points Raised
- Some participants describe superposition as the expression of a quantum state as a sum over multiple eigenstates of an observable, affecting measurement outcomes and uncertainty.
- Others note that maximum uncertainty occurs when all outcomes of an observable are equally likely, while minimum uncertainty occurs when the state is a single eigenstate.
- There is a discussion about whether the Heisenberg uncertainty relation is specific to position and momentum or applicable to general pairs of observables, with some suggesting it applies broadly.
- One participant questions if the superposition of eigenstates is equivalent to a mixed state, leading to clarification that a mixed state refers to a statistical ensemble of quantum states.
- Another participant connects the uncertainty principle to the concept of non-commutation of observables, suggesting an equivalence between the two concepts.
Areas of Agreement / Disagreement
Participants express varying views on the relationship between superposition and uncertainty, with some agreeing on the definitions while others raise questions about specific cases and interpretations. The discussion remains unresolved regarding the equivalence of superposition and mixed states.
Contextual Notes
Participants reference the need for clarity on definitions and the implications of non-commutation, indicating potential limitations in understanding the broader applications of the uncertainty principle.
- #2
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When the quantum state of a system |\psi\rangle is expressed as a sum over multiple eigenstates of an observable \hat{Q}, we say that the system is in a superposition of these different eigenstates (where each eigenstate corresponds to a particular possible measurement outcome of \hat{Q}). Depending on the particular superposition, you could be almost equally likely to measure any outcome of \hat{Q} (maximum uncertainty), or have one outcome's probability be much larger than the others (minimum uncertainty).
If |\psi\rangle happens to be in a single eigenstate of \hat{Q}, then when you measure \hat{Q}, you will get exactly the outcome associated to that eigenstate with 100 percent probability. In this case, the uncertainty in \hat{Q} is zero, since you know exactly what your measurement outcome would be.
However, just as |\psi\rangle can be expressed as a sum over the eigenstates of \hat{Q}, it can also be expressed as a different sum over the different eigenstates of another observable \hat{R}.
The uncertainty principle comes into play for observable pairs \hat{Q} and \hat{R} where there is no eigenstate of \hat{Q} that is also an eigenstate of \hat{R}. When that happens, there is no quantum state |\psi\rangle where you will be able to predict the measurement outcome of both \hat{Q} and \hat{R} with 100 percent certainty.
If |\psi\rangle happens to be in a single eigenstate of \hat{Q}, then when you measure \hat{Q}, you will get exactly the outcome associated to that eigenstate with 100 percent probability. In this case, the uncertainty in \hat{Q} is zero, since you know exactly what your measurement outcome would be.
However, just as |\psi\rangle can be expressed as a sum over the eigenstates of \hat{Q}, it can also be expressed as a different sum over the different eigenstates of another observable \hat{R}.
The uncertainty principle comes into play for observable pairs \hat{Q} and \hat{R} where there is no eigenstate of \hat{Q} that is also an eigenstate of \hat{R}. When that happens, there is no quantum state |\psi\rangle where you will be able to predict the measurement outcome of both \hat{Q} and \hat{R} with 100 percent certainty.
- #3
entropy1
- 1,232
- 72
Thank you very much for the explanation! 
To get this clear: is the Heisenberg uncertainty relation equal to this general form, or is it specific to impulse and position?
To get this clear: is the Heisenberg uncertainty relation equal to this general form, or is it specific to impulse and position?- #4
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entropy1 said:Thank you very much for the explanation!
The Heisenberg relation was originally expressed in terms of position and momentum, but it wasn't long before it was defined for general pairs of observables (where the position-momentum pair is a special case).
- #5
entropy1
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Is the superposition of eigenstates called a "mixed state"?
Is that non-commutation?
jfizzix said:
Is that non-commutation?
- #6
- 757
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Yes; that and non-commutation are equivalent.
Oops.. I answered the second question without looking at the first one. See radium's response for an answer to the first one.
Oops.. I answered the second question without looking at the first one. See radium's response for an answer to the first one.
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- #7
radium
Science Advisor
Education Advisor
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No, a mixed state is a statistical ensemble of several quantum states. Like an ensemble of spin 1/2 particles. The density matrix would be diagonal
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