Quantum Physics For Dummies, Revised Edition
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Finding the inverse of a large matrix often isn’t easy, so quantum physics calculations are sometimes limited to working with unitary operators, U, where the operator’s inverse is equal to its adjoint,

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(To find the adjoint of an operator, A, you find the transpose by interchanging the rows and columns, AT. Then take the complex conjugate,

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Note that the asterisk (*) symbol in the above equation means the complex conjugate. (A complex conjugate flips the sign connecting the real and imaginary parts of a complex number.)

This gives you the following equation:

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The product of two unitary operators, U and V, is also unitary because

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When you use unitary operators, kets and bras transform this way:

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And you can transform other operators using unitary operators like this:

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Note that the preceding equations also mean the following:

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Here are some properties of unitary transformations:

  • If an operator is Hermitian, then its unitary transformed version,

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  • is also Hermitian.

  • The eigenvalues of A and its unitary transformed version,

    image8.png
  • are the same.

  • Commutators that are equal to complex numbers are unchanged by unitary transformations:

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About This Article

This article is from the book:

About the book author:

Steven Holzner is an award-winning author of technical and science books (like Physics For Dummies and Differential Equations For Dummies). He graduated from MIT and did his PhD in physics at Cornell University, where he was on the teaching faculty for 10 years. He’s also been on the faculty of MIT. Steve also teaches corporate groups around the country.

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