This chapter should be read as a change of representation, not as a change of physics. Chapter 2 used wave functions because position and spatial boundary conditions were the natural language. Chapter 3 uses vectors and matrices because the Stern-Gerlach experiment presents a different kind of problem: a beam enters an apparatus and leaves through a small number of distinct channels.
The objective is to connect the postulates of quantum mechanics with a concrete laboratory story. A state becomes a ket, a physical question becomes an operator, possible experimental results become eigenvalues, and the relative intensities of outgoing beams become probabilities.
Matrix mechanics was introduced in the development of quantum theory as a way to compute transitions and discrete spectra without relying on a classical orbit picture. The formalism is especially natural for quantities with discrete values, such as angular momentum and spin. In that setting, the state is not most conveniently represented by a function of position, but by a list of amplitudes in a chosen basis.
The equivalence with wave mechanics is crucial. The two approaches give the same physical predictions; they simply organize the calculation differently. In wave mechanics a state may be written as \(\Psi(x,t)\). In matrix mechanics the same state can be represented by a vector of components:
The coefficients \(C_n\) carry the information needed to predict what happens when the system is tested in the basis \(\{|n\rangle\}\).
The chapter begins with Stern-Gerlach because it lets one see the entire formalism without hiding behind abstractions. Each step in the experiment motivates one mathematical object.
| Experimental idea | Formal object | Question answered |
|---|---|---|
| An incoming silver beam | State vector \(|\Psi\rangle\) | How is the system prepared? |
| A magnet oriented along an axis | Operator \(\hat\sigma_u\) | Which observable is being measured? |
| Two outgoing spots | Eigenvalues \(\pm 1\) | What results are possible? |
| Relative intensities | Born probabilities | How often does each result occur? |
| Blocking one beam | Projection/preparation | What state enters the next apparatus? |
| Changing magnet direction | Change of basis | How does the same state look in another basis? |
The book develops the chapter through definitions, examples and appendices. These pages keep that order, but make the connecting logic more explicit: after each equation, ask what part of the Stern-Gerlach story it describes. The goal is not to memorize isolated formulas, but to see how the apparatus forces the vector-matrix language to appear.
In Chapter 2, superposition was already present: a wave function could be expanded in stationary states. Chapter 3 abstracts that idea. A state is represented by a vector, and the expansion coefficients are projections onto a basis. This abstraction is useful because the basis need not be a position basis.
For spin, the two basis vectors are the two possible outcomes of a measurement along a chosen axis. The normalization condition already anticipates the probability interpretation: the outgoing beams must account for the whole incoming beam.
The chapter revisits the postulates from a new angle. Instead of writing everything with \(\Psi(x,t)\), it uses \(|\Psi(t)\rangle\), \(\hat A\), eigenvectors and inner products. The postulate for states becomes a statement about Hilbert-space vectors; the postulate for observables becomes a statement about Hermitian operators; the probability postulate becomes a statement about squared projections.
The notation introduced here is not limited to spin. The same ideas will return in angular momentum, perturbation theory, density operators and entanglement. Stern-Gerlach is a compact model in which all the essential mechanisms can be seen: basis choice, incompatible observables, state preparation and unitary time evolution.
This overview page is designed to orient later exercises on: Chapter map: why the vector-matrix language is natural for discrete quantum systems such as spin.