Quantum operators as coordinate systems

I think that some of us have better intuition about coordinate systems than we do about eigenvalues. Maybe a change in language will help make the concepts clearer.

• A quantity which you want to measure (let's use momentum ρ for example) is a coordinate system.
• Each potential value of the quantity (such as ρ = +1, ρ=+2 ) is its own axis in the coordinate system. This means that the coordinate system will have infinitely many dimensions when there are infinitely many possible values — but that's alright.
• A quantum object — such as a proton — is represented as a vector in this space. In general, the vector will not sit still, but will be swiveling around in this coordinate space as time goes on.
• A quantum state only has a well-defined measurement value (such as a well-defined value for its momentum) if it is parallel to one of the coordinate axes in the space. Otherwise, it has no well-defined value.
• The act of measuring, say, the momentum of a quantum state causes it to become parallel to one of the axes in the momentum's coordinate space. This gives it a well-defined momentum value, but will usually change the direction the vector is pointing in; measurement is destructive¹.
• Some physical quantities are compatible with each other; a particle can have a well-defined measurement for both quantities. This happens when two operators have coordinate systems whose axes are parallel to each other so that a vector can be simultaneously parallel to an axis in each system. Energy and momentum are compatible in this way.
• Some physical quantities are incompatible; their coordinate axes are in a skew relationship so that making a vector parallel to an axis in one system necessarily makes the vector not parallel to any axis in the other coordinate system. Position and momentum are incompatible quantities like this.

This intuitive way of describing quantum measurements can be made mathematically precise. In quantum mechanics:

• Measureable quantities are represented by linear operators (a generalization of a matrix).
• Quantum states are represented as vectors, which linear operators can operate on.
• Some linear operators have a special effect on certain vectors: they will change the size of those vectors without affecting their direction. Those specially-affected vectors are called eigenvectors of the linear operator, and the amount by which their size is changed is called the eigenvalue.
• When the linear operator represents a kind of measurement, an eigenvector is a quantum state that has well-defined measurement values—the well-defined measurement value is just the vector's eigenvalue.
• If a vector (quantum state) isn't an eigenvector, it doesn't have a well-defined measurement value.
• Eigenvectors with different eigenvalues are always perpendicular to each other. Thus, it makes sense to think of an opereator's eigenvectors as forming the axes of a coordinate system.
• Measuring a quantum state replaces that state with an eigenvector. In this way, measurement forces the vector to be aligned with one of the eigenvector axes.
• Although the choice of axis is apparently a random physical outcome, the probability distribution follows certain laws: the probability that a vector will be projected onto a particular axis depends on how far it extends in that axis's direction when it is measured.