Throughout this work, we have discussed the concept of measurement in quantum mechanics, emphasizing its crucial role, distinct from classical physics. Particularly, measuring an observable property of a quantum system influences the system’s state. The Stern-Gerlach experiment is frequently referenced to better understand this process. This experiment, which measures a particle’s spin and segregates it into specific spin states, demonstrates the fundamental principles of quantum measurement theory.
In quantum mechanical measurement, an observable quantity is represented by a self-adjoint operator. This operator is used to model the possible outcomes of the measurement process. Let us illustrate this concept with an example.
Before we proceed, a brief reminder is helpful: C² denotes a two-dimensional vector space over the field of complex numbers. Now, we move to the example.
The example is as follows:
- C² –> C² implies that the domain and codomain of a transformation are both within C².
- A = | 1 0 |
| 0 -1 |
The operator AA has two eigenvalues: +1 and −1. We will not discuss the process of finding eigenvalues here, but this will be covered in detail in the mathematics section. It is worth noting that eigenvalues of a matrix are derived from solving the characteristic equation for which the determinant equals zero.
These eigenvalues (in this case, +1 and −1) represent the real values obtainable during measurement. The eigenvectors corresponding to each eigenvalue indicate the state of the system after the measurement. Specifically:
- An eigenvalue of +1 implies that after the measurement, the system is in the state ∣↑⟩ (or ∣0⟩).
- An eigenvalue of −1−1 implies that after the measurement, the system is in the state ∣↓⟩∣↓⟩ (or ∣1⟩∣1⟩).
This corresponds to the two possible outcomes of the Stern-Gerlach experiment, where the measured spin component is either +1 or −1.
Now, we should discuss the superposition state and the measurement process. Let’s consider a pure quantum state that describes the system:
– ∣φ = 1/√2 |↑⟩ + 1/√2 |↓⟩
This state is a superposition state. A fundamental principle of quantum mechanics is that a quantum system can exist in multiple states simultaneously before measurement. The state ∣∣ϕ⟩ signifies that the particle is in both ∣↑⟩ and ∣↓⟩ states. During measurement, this superposition “collapses,” and the system transitions into one of the states associated with an eigenvalue of the operator A. This collapse provides definite information about the system’s state post-measurement. For example, if the measurement yields +1, the system transitions to the state ∣↑⟩ and remains in that state after the measurement.
Now, why did we write this superposition state and use the operator A? The superposition state relates to the condition of a spin being either up or down. The chosen operator A is used for measuring the spin component. Therefore:
- The selected operator must represent the physical quantity we wish to measure. In this case, the appropriate operator for measuring the spin component was used, and the corresponding superposition state was applied. Different operators would be used for different measurements.
From this example, we can state:
- In quantum mechanics, every measurement is associated with a specific operator, whose eigenvalues represent the measurement outcomes, and the eigenvectors represent the post-measurement state of the system.
- Here, the operator A determines the state transition of the system post-measurement and the eigenvalue to be observed. However, the operator itself does not execute the measurement; it models the potential results of the system.
Additional Note: While we will discuss this in more detail in the Quantum Computation and Mathematics sections, an initial explanation is worthwhile.
Eigenvalue: The real result obtainable during a measurement with a specific operator. For example, an operator measuring the z-component of spin can yield +1 or −1, which are its eigenvalues.
Eigenvector :
- for +1: ∣↑⟩ = ∣0⟩ ( 1 )
( 0 )
- for −1: ∣↓⟩ = ∣1⟩ ( 0 )
( 1 )