Can imaginary position operators explain real eigenvalues in quantum mechanics?

In summary, the conversation discusses the concept of operators in quantum mechanics giving real eigenvalues only when applied on the imaginary position, ix, rather than the normal position operator, x. This idea is brought up in the context of non-hermitian operators and there is a lack of literature on this topic. The conversation ends with the request for further information on complex operators and eigenvalues in quantum mechanics.
  • #1
SeM
Hello, some operators seem to "add up" and give real eigenvalues only if they are applied on the imaginary position, ix, rather than the normal position operator, x, in the integral :

\begin{equation}
\langle Bx, x\rangle
\end{equation}

when replaced by:\begin{equation}
\langle Bix, ix\rangle
\end{equation}

or\begin{equation}
\langle Bix, x\rangle
\end{equation}

So using (2) or even (3) I get real (but negative) eigenvalues, instead of complex eigenvalues. I have not found any literature on such an absurd thing, as the imaginary position, however, being the only answer to these operators, I am wondering if anyone can point to some further literature on complex operators and complex eigenvalues in QM and whether (2) and (3) make any sense at all

Thanks
 
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  • #2
What's ##B##? Without clear definitions, I cannot make any sense of the expressions above. Where does this idea of complex/imaginary position eigenvalues come from?
 
  • #3
It's an idea that comes from a calculation that sums up only if I use an imaginary position operator. Apparently, it is so non-heard of that I will leave it as it is. It has to do with non-hermiticity, where B (not hermitian) only gives a real value on that integral if the position is imaginary.

Thanks!
 

1. What is an Imaginary Position Operator?

An Imaginary Position Operator is a mathematical operator used in quantum mechanics to represent the imaginary component of a particle's position. It is often denoted as iX, where X is the position operator.

2. How is the Imaginary Position Operator used in quantum mechanics?

The Imaginary Position Operator is used to describe the quantum state of a particle, which is a mathematical representation of its position and momentum. It is an important tool in calculating the probability of a particle being at a certain position in space.

3. What is the difference between the Imaginary Position Operator and the Position Operator?

The main difference between the Imaginary Position Operator and the Position Operator is that the former represents the imaginary component of a particle's position, while the latter represents the real component. Both are used in quantum mechanics to describe the position of a particle, but they operate on different mathematical principles.

4. How is the Imaginary Position Operator related to the Uncertainty Principle?

The Imaginary Position Operator plays a key role in the Uncertainty Principle, which states that the more precisely we know a particle's position, the less precisely we know its momentum, and vice versa. The Imaginary Position Operator is used to calculate the uncertainty in a particle's position.

5. Can the Imaginary Position Operator be physically measured?

No, the Imaginary Position Operator is a mathematical concept and cannot be physically measured. It is a tool used to describe the position of a particle in quantum mechanics, but it does not have a physical counterpart. Only the real component of a particle's position can be measured.

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