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dimensionless
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What is the Modulus of an Eigenvalue?
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In summary, the modulus of a real number is its absolute value, and in quantum mechanics, the eigenvalues are assumed to be real. The modulus of a complex number is the square root of the sum of the squares of its real and imaginary parts. This can also be represented as a vector in the complex plane. This idea was first introduced by Argand, but Feynman also referred to complex amplitudes as "little arrows" in his book QED. In quantum mechanics, every observable has a hermitian operator representation, ensuring that they have real eigenvalues. This means that the modulus of a complex number can change sign if its eigenvalue is negative.
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LeonhardEuler
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The modulus of a real number is its absolute value. Since this is posted under quantum mechanics, I am assuming the the eigenvalue is real. In a more general case, though, the modulus of a complex number, a + bi, is [itex]\sqrt{a^2+b^2}[/itex].
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selfAdjoint
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Yes, if you regard a complex number as a vector in the plane (Feynmann;s "little arrows") then its modulus is its length. This obviously agrees with LeonhardEuler's algebraic definition.
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masudr
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I'd credit the arrows to Argand, not Feynman...
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selfAdjoint
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masudr said:
Absolutely!
I wasn't giving him credit for the idea, but in his little book QED he refers to the complex amplitudes on his paths as little arrows. I always thought that was both sharp and funny.- #6
vroom
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If u r talking of QM. Then this question appears meaningless to me.
In QM, every observable has got a hermitian operator representation. By the mathematics of hemitians we know they always have real eigenvalues.
so a mod amounts to change of sign if the eigval is -ve
In QM, every observable has got a hermitian operator representation. By the mathematics of hemitians we know they always have real eigenvalues.
so a mod amounts to change of sign if the eigval is -ve
1. What is the Modulus of an Eigenvalue?
The modulus of an eigenvalue is the absolute value or magnitude of the eigenvalue. It represents the distance of the eigenvalue from the origin on the complex plane.
2. How is the Modulus of an Eigenvalue calculated?
The modulus of an eigenvalue is calculated by taking the square root of the sum of the squares of the real and imaginary parts of the eigenvalue.
3. What is the significance of the Modulus of an Eigenvalue?
The modulus of an eigenvalue is an important property in linear algebra and is used to determine the stability and behavior of a system. It also helps in determining the properties of a matrix, such as whether it is invertible or singular.
4. Can the Modulus of an Eigenvalue be negative?
No, the modulus of an eigenvalue is always a positive value. This is because it represents the distance from the origin on the complex plane and distance is always a positive quantity.
5. How does the Modulus of an Eigenvalue relate to the trace of a matrix?
The modulus of an eigenvalue is related to the trace of a matrix through the characteristic equation. The trace of a matrix is equal to the sum of its eigenvalues, and the characteristic equation helps in finding the eigenvalues of a matrix.
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