Why are the probabilities cos²(θ/2) and sin²(θ/2), and not just cos²θ?

The half-angle formula arises from the quantum mechanical property of spin-½ particles. Their spin states transform under rotations like objects with a 720° periodicity, not 360°. The probability is the squared magnitude of the probability amplitude, which for spin-½ involves cos(θ/2) and sin(θ/2). This is a direct signature of the particle's intrinsic angular momentum and its representation by two-component spinors.

What happens if I block one of the beams exiting the first SG device?

Blocking, say, the |-z⟩ beam creates a pure state of |+z⟩ particles entering the second device. This is crucial for state preparation. If you did not block it, you would have a statistical mixture of |+z⟩ and |-z⟩ particles, and the results at SG₂ would be a different weighted average of probabilities, not simply cos²(θ/2). The simulator assumes this filtering is done.

Is this only about electron spin, or does it apply to other things?

The same mathematical formalism applies to any two-level quantum system, making it a universal prototype. Beyond electron spin, it models photon polarization (where θ/2 becomes θ), nuclear spin in NMR/MRI, and artificial qubits in quantum computing. The Stern–Gerlach setup is the historical paradigm for understanding measurement in such systems.

Does the simulator show the actual magnetic field gradients and particle deflection?

No, this is a key simplification. The simulator abstracts away the classical magnetic force (F = -∇(μ·B)) that causes spatial splitting in a real SG magnet. It focuses solely on the quantum probabilistic outcomes after the state preparation and measurement stages, which is the core conceptual lesson for understanding quantum mechanics.

Sequential Stern–Gerlach

Two SG devices: P(up on SG₂) = cos²(θ/2) or sin²(θ/2) after |±z⟩ filter.

About this model

Sequential Stern–Gerlach experiments are a foundational tool in quantum mechanics for illustrating the quantization of angular momentum and the probabilistic nature of measurement. This simulator visualizes the passage of spin-½ particles, such as electrons or silver atoms, through two Stern–Gerlach (SG) devices in sequence. The first device, SG₁, acts as a polarizer or filter, preparing particles in a definite spin state along one axis (e.g., |+z⟩). The second device, SG₂, is oriented at an angle θ relative to the first, measuring the spin component along a new axis. The core physics is governed by the quantum mechanical rule for projecting one quantized spin state onto another. For a particle prepared in |+z⟩, the probability to be measured as |+θ⟩ in the second device is P(+) = cos²(θ/2), while the probability for |-θ⟩ is P(-) = sin²(θ/2). These probabilities derive from the inner product (overlap) of the respective quantum state vectors. The simulator makes key simplifications: it assumes an ideal, 100% efficient first filter, neglects any spatial wave packet evolution or magnetic field details, and treats the particles as non-interacting. By interacting with the simulator, students directly explore the non-classical consequences of sequential measurements, observe how probability varies continuously with the relative angle θ even though individual outcomes are discrete, and reinforce their understanding of state preparation, measurement collapse, and the fundamental distinction between quantum superposition and statistical mixtures.

Who it's for: Undergraduate students in quantum mechanics, modern physics, or physical chemistry courses learning about spin, quantum measurement, and foundational experiments.

Key terms

Stern–Gerlach experiment
Spin-½
Quantum measurement
State preparation
Probability amplitude
Projection probability
Quantum superposition
Quantization of angular momentum

How it works

Sequential projection illustrates quantum measurement bases: intermediate paths matter, unlike additive classical magnets.

Frequently asked questions

Why are the probabilities cos²(θ/2) and sin²(θ/2), and not just cos²θ?: The half-angle formula arises from the quantum mechanical property of spin-½ particles. Their spin states transform under rotations like objects with a 720° periodicity, not 360°. The probability is the squared magnitude of the probability amplitude, which for spin-½ involves cos(θ/2) and sin(θ/2). This is a direct signature of the particle's intrinsic angular momentum and its representation by two-component spinors.
What happens if I block one of the beams exiting the first SG device?: Blocking, say, the |-z⟩ beam creates a pure state of |+z⟩ particles entering the second device. This is crucial for state preparation. If you did not block it, you would have a statistical mixture of |+z⟩ and |-z⟩ particles, and the results at SG₂ would be a different weighted average of probabilities, not simply cos²(θ/2). The simulator assumes this filtering is done.
Is this only about electron spin, or does it apply to other things?: The same mathematical formalism applies to any two-level quantum system, making it a universal prototype. Beyond electron spin, it models photon polarization (where θ/2 becomes θ), nuclear spin in NMR/MRI, and artificial qubits in quantum computing. The Stern–Gerlach setup is the historical paradigm for understanding measurement in such systems.
Does the simulator show the actual magnetic field gradients and particle deflection?: No, this is a key simplification. The simulator abstracts away the classical magnetic force (F = -∇(μ·B)) that causes spatial splitting in a real SG magnet. It focuses solely on the quantum probabilistic outcomes after the state preparation and measurement stages, which is the core conceptual lesson for understanding quantum mechanics.

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About this model

How it works

Frequently asked questions

Related simulators

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Sequential Stern–Gerlach

About this model

How it works

Frequently asked questions