Undergrad Operation on complex conjugate

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SUMMARY

The discussion centers on the operation of sandwiching operators in quantum mechanics, specifically why operators act on wavefunctions rather than their complex conjugates. It highlights that while one could define inner products differently, the results remain consistent. A critical point raised is that allowing operations on both wavefunctions and their conjugates could lead to violations of causality, such as enabling faster-than-light communication.

PREREQUISITES
  • Understanding of quantum mechanics principles
  • Familiarity with wavefunctions and complex conjugates
  • Knowledge of inner product spaces in mathematics
  • Basic grasp of causality in physics
NEXT STEPS
  • Research the mathematical framework of inner product spaces
  • Explore the implications of complex conjugates in quantum mechanics
  • Study causality and its importance in relativistic physics
  • Examine the role of operators in quantum mechanics
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Physicists, quantum mechanics students, and mathematicians interested in the foundational principles of quantum theory and the implications of operator actions on wavefunctions.

Thejas15101998
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Why do we sandwich operators in quantum mechanics in such a way that the operator acts on the wavefunction and not on its complex conjugate?
 
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Thejas15101998 said:
Why do we sandwich operators in quantum mechanics in such a way that the operator acts on the wavefunction and not on its complex conjugate?

You could define things the other way round. In fact, using the "mathematician's" definition of an inner product would probably have things that way round. All the results would be the same.

See, for example:

https://en.wikipedia.org/wiki/Inner_product_space
 
As a follow-up on what PeroK said, note that it is important that you can only do one. If you can operate on a wavefunction and also on that wavefunction's conjugate (or, equivalently, if you're given an operation that can conjugate the wavefunction), it becomes possible to send information faster than light.
 

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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