Explain the measurement axiom in quantum mechanics and how it affects the state of a system after measurement.
The measurement axiom is a fundamental concept in quantum mechanics that describes the effect of measurement on the state of a quantum system. It states that when a measurement is performed on a quantum system, the system will collapse into one of the eigenstates of the observable being measured, with the probability of each outcome determined by the coefficients of the system's state vector in the corresponding eigenbasis.
To understand the measurement axiom, let's consider a quantum system described by a state vector |ψ⟩ in a K-level system. The state vector |ψ⟩ represents the quantum state of the system, and it can be written as a linear combination of the basis vectors |k⟩, where k ranges from 1 to K. In the bra-ket notation, we can express the state vector as |ψ⟩ = ∑c_k |k⟩, where c_k are complex coefficients.
When a measurement is performed on the system, it is associated with an observable, which is a Hermitian operator. The eigenstates of the observable form a complete orthonormal basis for the system. Let's denote the eigenstates of the observable as |e_i⟩, where i ranges from 1 to K. The eigenvalues associated with these eigenstates are denoted as λ_i.
According to the measurement axiom, when a measurement is made on the system, the state vector |ψ⟩ collapses into one of the eigenstates |e_i⟩ with the probability given by the squared modulus of the coefficient of the state vector in the corresponding eigenbasis. In other words, the probability of obtaining the measurement outcome corresponding to the eigenstate |e_i⟩ is given by |c_i|^2.
After the measurement, the system will be in the eigenstate |e_i⟩ associated with the measurement outcome. This is known as the collapse of the wavefunction. The state vector |ψ⟩ is now replaced by the eigenstate |e_i⟩.
Let's illustrate this with an example. Consider a qubit, which is a two-level quantum system. The state vector of the qubit can be written as |ψ⟩ = c_0 |0⟩ + c_1 |1⟩, where |0⟩ and |1⟩ are the basis states of the qubit and c_0 and c_1 are complex coefficients. Suppose we measure the qubit in the computational basis, which is the eigenbasis of the Pauli-Z operator. The eigenstates of the Pauli-Z operator are |0⟩ and |1⟩, with eigenvalues +1 and -1, respectively.
If the measurement outcome is +1 (corresponding to the eigenstate |0⟩), the state of the qubit after measurement will collapse to |0⟩. Similarly, if the measurement outcome is -1 (corresponding to the eigenstate |1⟩), the state of the qubit after measurement will collapse to |1⟩. The probability of obtaining each outcome is given by |c_0|^2 and |c_1|^2, respectively.
The measurement axiom in quantum mechanics states that when a measurement is performed on a quantum system, the system collapses into one of the eigenstates of the observable being measured, with the probability of each outcome determined by the squared modulus of the coefficient of the state vector in the corresponding eigenbasis. This collapse of the wavefunction is a fundamental aspect of quantum mechanics and has important implications for the behavior of quantum systems.
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