Find Complex Conjugate of u_j in Adjoint of Equation

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SUMMARY

The discussion centers on the mathematical operations involving the adjoint of an equation in quantum mechanics, specifically referencing Shankar's work. The equations U|u_i>=u_i|u_i> and U|u_j>=u_j|u_j> are analyzed, leading to the conclusion that {u_j}^* represents the complex conjugate of u_j. The adjoint operation and the inner product are crucial in deriving the relationship between these states. The clarity achieved in understanding the notation confirms that u* is indeed the complex conjugate of u.

PREREQUISITES
  • Understanding of quantum mechanics notation and principles
  • Familiarity with adjoint operators in linear algebra
  • Knowledge of inner product spaces
  • Basic concepts of complex numbers and their conjugates
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  • Study the properties of adjoint operators in quantum mechanics
  • Learn about inner product spaces and their applications in quantum theory
  • Explore Shankar's "Principles of Quantum Mechanics" for deeper insights
  • Research complex conjugates and their significance in quantum state representations
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Students and professionals in physics, particularly those studying quantum mechanics, as well as mathematicians interested in linear algebra applications in physics.

cragar
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This is not a homework problem, just trying to figure out what shankar did.
[itex]U|u_i>=u_i|u_i>[/itex]
[itex]U|u_j>=u_j|u_j>[/itex]
Ok so we have these 2 equations and then shankars says to take the adjoint
of the second equation and dot each side with the corresponding side of the first equation,
we get
[itex]<u_j|U^tU|u_i=u_i{u_j}^*<u_j|u_i>[/itex]
is the [itex]{u_j}^*[/itex] just the complex conjugate of [itex]u_j[/itex] ?
Now that I typed it out it seems more clear I originally thought there was an extra factor.
 
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Yes, u* is the complex conjugate of u.
 

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