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ehrenfest
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This is probably really obvious but can someone explain to me why adjiont( U(t)) * U(t) = I where U(t) is a propagator in QM and I is the identity.
ehrenfest said:This is probably really obvious but can someone explain to me why adjiont( U(t)) * U(t) = I where U(t) is a propagator in QM and I is the identity.
The term "propagator" comes from the idea that U(t) represents the evolution of a quantum state over time. It is a mathematical operator that describes how a system evolves from one state to another at a specific time t.
The notation "adj(U(t))" refers to the adjoint or Hermitian conjugate of the operator U(t). In simple terms, it is the complex conjugate of the transpose of U(t).
This is a fundamental property of unitary operators in Quantum Mechanics. The adjoint of an operator is defined in such a way that when multiplied with the original operator, it produces the identity operator. This is important because unitary operators preserve the normalization of quantum states.
The equation U(t) * adj(U(t)) = I ensures that the probability of finding a particle in any state remains the same throughout its evolution. This is known as the conservation of probability, and it is a fundamental principle in Quantum Mechanics.
Yes, this equation is a general property of unitary operators and can be applied to any quantum system. It is a fundamental principle that governs the evolution of quantum states in time.