# Exponential of an operator into bra-ket notation

• Bravus
In summary, the student is looking for a way to get started with the bra-ket notation and is struggling with the concept. He is also confused about what e-iA|∑anxn> means.
Bravus

## Homework Statement

The question is to evaluate the expression e^-iA, where A is a Hermitian operator whose eigenvalues are known (but not given) using bra-ket algebra.

See above.

## The Attempt at a Solution

I have been looking around, reading the textbook and course notes, checking the web and so on. I have learned that an expression in this form is a 'unitary operator', and what that means.

I'm pretty comfortable with the bra-ket notation, but I'm struggling just to *get into it*. ;-)

If you can help me find a 'way in' to get started, that'd be great: at this point I'm still just kinda staring blankly at it, despite all my efforts so far.

Oh, yeah, I overhead another student mention using a Taylor series. I get how to convert the exponential to a Taylor series, but I'm not so sure how it helps... and the other student might be wrong.

Welcome to PF!

Hi Bravus! Welcome to PF!

(try using the X2 button just above the Reply box )
Bravus said:
The question is to evaluate the expression e^-iA, where A is a Hermitian operator whose eigenvalues are known (but not given) using bra-ket algebra.

I think they mean if A|xn> = λn |xn>,

then what is e-iA|∑anxn> ?

Thanks for the tip on the nicenesses allowed by the forum software - so nice after spending time answering math and physics questions on 'Yahoo Answers' and having to struggle with all-text!

Not sure the answer is giving me a way in, though: maybe I'm just thick. The question definitely just says 'evaluate e-iA'.

Hi Bravus!

(just got up :zzz:)

Bravus said:
The question definitely just says 'evaluate e-iA'.

but anyway, what is e-iA|∑anxn> ?

Thanks, and the hint is definitely a handy one. It's Mr Taylor and his Series that really gets the job done in this instance, though...

## 1. What is the definition of an exponential of an operator in bra-ket notation?

The exponential of an operator in bra-ket notation is a mathematical expression that represents the result of repeatedly applying the operator to a state vector. It is denoted as eĤ where e is the base of the natural logarithm, and Ĥ is the operator.

## 2. How is the exponential of an operator calculated in bra-ket notation?

The exponential of an operator in bra-ket notation is calculated using the Taylor series expansion of the operator. This involves breaking down the operator into a sum of its powers and applying each power to the state vector. The resulting series is then summed to obtain the final exponential expression.

## 3. What are the properties of the exponential of an operator in bra-ket notation?

The exponential of an operator in bra-ket notation has several important properties, including linearity, unitarity, and Hermiticity. It also satisfies the commutation relation [eĤ, A] = 0, where A is any other operator.

## 4. How is the exponential of an operator used in quantum mechanics?

The exponential of an operator is a useful tool in quantum mechanics for calculating the time evolution of a quantum system. It is used to solve the Schrödinger equation and to determine the probability amplitudes for different states of a quantum system at different times.

## 5. Can the exponential of an operator be applied to any type of operator in bra-ket notation?

Yes, the exponential of an operator can be applied to any type of operator in bra-ket notation, including both Hermitian and non-Hermitian operators. However, for non-Hermitian operators, the exponential may not represent a physical observable and should be used with caution.

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