Undergrad Uncertainty relation derivation

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SUMMARY

The discussion centers on the uncertainty relation derivation in quantum mechanics, specifically addressing the commutation of two operators. The lecturer asserts that if two operators commute, they can be measured simultaneously; if they do not, the uncertainty relation applies, expressed as 2ΔqΔs ≥ abs(<[Q^,S^]>). Participants seek a formal derivation of this relation and explore its implications on measuring momentum and position simultaneously.

PREREQUISITES
  • Understanding of quantum mechanics principles
  • Familiarity with operator algebra in quantum physics
  • Knowledge of commutation relations
  • Basic grasp of the uncertainty principle
NEXT STEPS
  • Research the derivation of the uncertainty relation in quantum mechanics
  • Study the implications of commutation relations on measurement
  • Explore the mathematical framework of operator theory in quantum mechanics
  • Examine case studies involving simultaneous measurements of momentum and position
USEFUL FOR

Students of quantum mechanics, physicists interested in operator theory, and anyone seeking to deepen their understanding of the uncertainty principle and its applications in measurement theory.

Somali_Physicist
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Hey guys , my lecturer introduced a new concept with reference to the commutation of two operators.He claimed that if two commutators commute then they can be simultaneously measured.I can clearly see how this works.He then went on and state if they don't commute they can't simultaneously be measured.However he went further and claimed if they cannot commute then the uncertainty relation between the operators values is:

2ΔqΔs ≥ abs(<[Q^,S^]>)

Does anyone have a derivation for this bizaare relation.

Finally , if we measure the y momentum of an object , can we also meausre the x and z position?
 
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Thread 'Two equivalent statements of time reversal symmetric Hamiltonian'

Time reversal invariant Hamiltonians must satisfy ##[H,\Theta]=0## where ##\Theta## is time reversal operator. However, in some texts (for example see Many-body Quantum Theory in Condensed Matter Physics an introduction, HENRIK BRUUS and KARSTEN FLENSBERG, Corrected version: 14 January 2016, section 7.1.4) the time reversal invariant condition is introduced as ##H=H^*##. How these two conditions are identical?
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