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meteorologist1
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Hi, could anyone tell me how one would show that the expectation value of a anti-Hermitian operator is a pure imaginary number? Thanks.
HallsofIvy said:By the way, "expectation" is a noun. "Expected" is an adjective.
The "expected value" is the "expectation".
The expectation value of an anti-Hermitian operator is a mathematical concept in quantum mechanics that represents the average measurement of a physical quantity in a quantum system. It is calculated by taking the inner product of the state vector with the operator and the state vector again.
The expectation value of an anti-Hermitian operator is purely imaginary, while the expectation value of a Hermitian operator is a real number. This is because anti-Hermitian operators represent anti-symmetric or imaginary physical quantities, while Hermitian operators represent symmetric or real physical quantities.
The expectation value of an anti-Hermitian operator is significant because it allows us to predict the average measurement of a physical quantity in a quantum system. It also plays a crucial role in determining the evolution of quantum states over time, as it is related to the Hamiltonian operator which governs the time evolution of quantum systems.
Yes, the expectation value of an anti-Hermitian operator can be negative. This is because anti-Hermitian operators can represent imaginary or anti-symmetric physical quantities, which can have negative values.
The expectation value of an anti-Hermitian operator is related to uncertainty in quantum mechanics through the Heisenberg uncertainty principle. This principle states that the product of the uncertainties in two non-commuting operators, such as an anti-Hermitian operator and a Hermitian operator, must be greater than or equal to the absolute value of their commutator. This means that the expectation value of an anti-Hermitian operator can give us information about the uncertainty in a corresponding physical quantity.