Up to this point, we have discussed the Stern-Gerlach experiment in several sections. This experiment allows us to observe one of the fundamental principles of quantum mechanics: Spin Quantization, which plays a significant role in the representation of quantum states.

By the expression “Spin Quantization,” we mean that a particle’s quantum property, spin, can only take on specific discrete values. For example, in spin-1/2 particles, the spin components (when measured along the direction of a magnetic field) can only have two distinct values: spin “up” or “down.” These discrete values emerge not as continuous orientations of a classical rotating object but as specific quantized (discrete) states. Spin quantization was first observed in the Stern-Gerlach experiment, where the silver atoms deflected in two discrete directions in a magnetic field, demonstrating that their spin is quantized. This property is one of the fundamental principles of quantum mechanics, where the intrinsic angular momentum of particles, known as spin, can only be measured at certain quantized values.

We have previously explained that the Stern-Gerlach experiment particularly reveals that the measurement results of a quantum particle’s spin components are framed within specific probabilities. From this, we can infer that pure quantum states are probabilistically dependent on measurements and that they will manifest with a certain probability. These probabilities are based on the components of the vector representing a quantum state.

For instance, considering a spin-1/2 particle, the particle’s up spin ∣↑⟩ and down spin ∣↓⟩ states form two separate basis states. These basis states can be expressed as a linear combination of any quantum state, that is,

ψ = α↑ ∣↑⟩ + α↓ ∣↓⟩

Here, α↑​ and α↓ are complex coefficients (amplitudes) proportional to the probabilities of measuring the particle in the up or down spin states, respectively. The absolute squares of these coefficients, ∣α↑∣² and ∣α↓∣², determine the probabilities of measuring the spin as up or down.

This representation shows that quantum states must be expressed in a vector space (also known as Hilbert space). The definition of the inner product in such a space is crucial because it allows concepts such as the norm (unit length) and orthogonality of quantum states to gain meaning. The inner product also enables the determination of the angle between states, and consequently, their probabilities; thus, it plays a fundamental role in the representation of quantum states as vectors.

In this context, the results of the Stern-Gerlach experiment illustrate how the vectors representing pure states in quantum mechanics reflect the probabilities of measurement outcomes through their inner products. This inner product also allows us to choose the norm of quantum vectors to be one, enabling us to express pure states as unit vectors. Moreover, the orthogonality of two states indicates that the states corresponding to different measurement outcomes are independent (i.e., they do not influence each other during measurement). This representation in the Stern-Gerlach experiment contributes to the mathematical framework of quantum mechanics, highlighting the necessity of representing pure quantum states with vectors in Hilbert space.