What happens to the state of a system after measuring an observable with repeated eigenvalues?
When measuring an observable with repeated eigenvalues in a quantum system, the state of the system undergoes a collapse into one of the corresponding eigenstates. To understand this phenomenon, we need to consider the mathematical framework of quantum mechanics and the concept of observables.
In quantum mechanics, observables are represented by Hermitian operators. These operators have a set of eigenvalues and corresponding eigenvectors. The eigenvalues represent the possible outcomes of a measurement, while the eigenvectors represent the states in which the system can be found after the measurement.
When an observable has repeated eigenvalues, it means that there are multiple eigenvectors associated with the same eigenvalue. Let's consider a simple example to illustrate this concept. Suppose we have a system with a spin-1/2 particle, and we want to measure its z-component of spin. The observable in this case is the spin operator along the z-axis, denoted by Sz. The eigenvalues of Sz are +ħ/2 and -ħ/2, representing the possible outcomes of the measurement.
Now, let's assume that the system is initially in a superposition state given by |ψ⟩ = α|+⟩ + β|-⟩, where |+⟩ and |-⟩ are the eigenvectors corresponding to the eigenvalues +ħ/2 and -ħ/2, respectively. Here, α and β are complex probability amplitudes that satisfy the normalization condition |α|^2 + |β|^2 = 1.
When we measure the z-component of spin, the system collapses into one of the eigenstates. The probability of obtaining the eigenvalue +ħ/2 is given by |α|^2, and the probability of obtaining the eigenvalue -ħ/2 is given by |β|^2. After the measurement, the state of the system becomes either |+⟩ or |-⟩, depending on the outcome of the measurement.
It is important to note that the act of measurement disturbs the quantum system, causing the collapse of the wavefunction. Prior to the measurement, the system was in a superposition state, but after the measurement, it is in a definite state corresponding to the measured eigenvalue. This collapse is a fundamental aspect of quantum mechanics and is often referred to as the "measurement problem."
To summarize, when measuring an observable with repeated eigenvalues, the state of the system collapses into one of the corresponding eigenstates. The probability of obtaining a particular eigenvalue is determined by the squared magnitudes of the complex probability amplitudes associated with the corresponding eigenvectors. This collapse is a consequence of the measurement process and is a key feature of quantum mechanics.
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