Proving U=eiA is Unitary: Exploring -1 as Exponent and Inverse

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In summary: More generally, if operators/matrices ##A## and ##B## commute, then you can show that:$$e^Ae^B = e^{A+B}$$It shouldn't be hard to find a proof online if you don't want to work it out for yourself.
  • #1
dyn
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Hi
If A is a Hermitian operator then U = eiA is a unitary operator.
To prove this we take the Hermitian conjugate of U

U+ =
e-iA = (eiA)-1 (1)
U+ = U-1 (2)

My question is - In line (1) , -1 is used as an exponent or power while in line (2) , -1 is used to refer to the inverse of a matrix. Are these not 2 different uses of -1 ?
Thanks
 
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  • #2
dyn said:
My question is - In line (1) , -1 is used as an exponent or power while in line (2) , -1 is used to refer to the inverse of a matrix. Are these not 2 different uses of -1 ?
In general, ##X^{-1}## denotes the multiplicative inverse of ##X##, where ##X## is some mathematical object. In this case, you have to justify the index rule ##e^{-iA} = (e^{iA})^{-1}## for a (Hermitian) operator ##A##.
 
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PeroK said:
In general, ##X^{-1}## denotes the multiplicative inverse of ##X##, where ##X## is some mathematical object. In this case, you have to justify the index rule ##e^{-iA} = (e^{iA})^{-1}## for a (Hermitian) operator ##A##.
Thanks. So the -1 in the equation you wrote is not an exponent ; it is the inverse of eiA ? How do i justify that ?
 
  • #4
dyn said:
Thanks. So the -1 in the equation you wrote is not an exponent ; it is the inverse of eiA ? How do i justify that ?
You prove it, using the definition of ##e^A##.
 
  • #5
I know the definition of eA as an infinite power series but i don't know how to get the inverse of that
 
  • #6
dyn said:
I know the definition of eA as an infinite power series but i don't know how to get the inverse of that
More generally, if operators/matrices ##A## and ##B## commute, then you can show that:$$e^Ae^B = e^{A+B}$$It shouldn't be hard to find a proof online if you don't want to work it out for yourself.
 
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It's not that i don't work it out myself ; i don't know how to !
 
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  • #8
I found it online. Thanks
 

1. What is the definition of a unitary matrix?

A unitary matrix is a square matrix whose conjugate transpose is equal to its inverse. In other words, multiplying a unitary matrix by its conjugate transpose results in the identity matrix.

2. How do you prove that U=eiA is unitary?

To prove that U=eiA is unitary, we need to show that U multiplied by its conjugate transpose equals the identity matrix. This can be done by using the properties of complex numbers and the exponential function, and by showing that the inverse of U is equal to its conjugate transpose.

3. Why is -1 used as the exponent in U=eiA?

The use of -1 as the exponent in U=eiA is based on the properties of the exponential function. When the exponent is -1, the exponential function becomes its own inverse. This allows us to easily show that U multiplied by its conjugate transpose is equal to the identity matrix, which is a key requirement for a unitary matrix.

4. How does exploring -1 as the exponent in U=eiA help in understanding unitary matrices?

Exploring -1 as the exponent in U=eiA allows us to understand the fundamental properties of unitary matrices. It helps us see how the use of complex numbers and the exponential function can result in a matrix that is its own inverse, and how this relates to the definition of a unitary matrix.

5. Can U=eiA be used to solve problems in quantum mechanics?

Yes, U=eiA has many applications in quantum mechanics. It is commonly used to represent unitary transformations in quantum systems, where the exponential function can be used to describe the time evolution of a quantum state. It also plays a key role in the study of quantum gates and quantum computing.

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