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Andrew
Andrew
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ADDENDUM For completeness, I'll add that in addition to $\hat{S}_z$, two other important operators that measure properties of electrons that can be known at the same time as energy is known are $\hat{L}^2$, the operator for the magnitude of total (quantum) angular momentum and $\hat{L}_z$, the z-component of that angular momentum. Thus, we can theoretically have an electron for which all four properties are fixed at one time, and we define the state described by the four eigenvalues by representing those eigenvalues with the "quantum numbers" (simplified forms of the eigenvalues) with which you are probably familiar:

n is the quantum number which represents the eigenvalues of the energy operator $\hat{H}$ (ie the Hamiltonian, which we associate with a specific shell)

l is the quantum number which represents the eigenvalues of the operator $\hat{L}^2$ (which we equate with s,p, d etc orbitals)

ml is the quantum number which represents the eigenvalues of the operator $\hat{L}_z$ (which we equate with the axial orientation of orbitals)

ms is the quantum number which represents the eigenvalues of the operator $\hat{S}_z$ (which we equate with the above-described spin orientation)

ADDENDUM For completeness, I'll add that in addition to $\hat{S}_z$, two other important operators that measure properties of electrons that can be known at the same time as energy is known are $\hat{L}^2$, the operator for the magnitude of total (quantum) angular momentum and $\hat{L}_z$, the z-component of that angular momentum. Thus, we can theoretically have an electron for which all four properties are fixed at one time, and we define the state described by the four eigenvalues by representing those eigenvalues with the "quantum numbers" (simplified forms of the eigenvalues) with which you are probably familiar:

n is the quantum number which represents the eigenvalues of the energy operator $\hat{H}$ (ie the Hamiltonian, which we associate with a specific shell)

l is the quantum number which represents the eigenvalues of the operator $\hat{L}^2$ (which we equate with s,p, d etc orbitals)

ml is the quantum number which represents the eigenvalues of the operator $\hat{L}_z$ (which we equate with the axial orientation of orbitals)

ms is the quantum number which represents the eigenvalues of the operator $\hat{S}_z$ (which we equate with the above-described spin orientation)

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Andrew
Andrew
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In QM, properties that can be measured (such as energy, spin, position, etc) are each associated with an "operator". Every observation will always return an eigenvalue for the relevant operator. Eigen values are values for $\lambda$ for which the equation $$\hat{O}\Psi = \lambda\Psi$$ is true, where $\hat{O}$ is the operator of interest and $\Psi$ is the wave function of the entity of interest.

For example, let's consider electron "spin", by which we typically mean specifically the component of the spin in the z-direction, given by the operator $\hat{S}_z$. For the electron, this operator has two possible eigenvalues, which we abbreviate as $+1/2$ and $-1/2$. An electron having one of these values of spin is described as being in the eigenstate (or more simply just "state") associated with the given eigenvalue.

So we can say that having $+1/2$ spin and having $-1/2$ spin are two possible states of the electron.

What makes QM different from classical mechanics is that measuring an electron to have $+1/2$ spin does not mean that it was necessarily in the $+1/2$ spin state prior to our taking the measurement. An electron can (and often does) exist in a super-position of states, meaning that it simultaneously has both $+1/2$ and $-1/2$ spin, but our measurement causes it to collapse into one state or the other.

As you described in your question, some pairs of operators have the property that you cannot measure a defined value for both at the same time. Position and momentum are the most famous pair, since they are the example for the Heisenberg uncertainty principle, but many other pairs behave similarly. Energy and position are one such pair. So if we measure energy precisely, we cannot know where the electron with that energy exists. Since energy levels are the basis of the Schrodinger equation and define much of the behavior of atoms, chemists tend to treat electrons as if they are in a state in which the energy has been measured and other properties are unknown.

To address your point of confusion - "wave" and "particle" are descriptions of the behavior of electrons, most useful when understanding the concept of positional uncertainty, since "position" is most associated with particle behavior. When we learn about the Heisenberg uncertainty, we learn about experiments in which electrons behave as waves when their position has not been measured and like particles when their position has been measured, so it's easy to see why you associate wave and particle behavior with velocity and position, respectively, but those are just convenient examples to show the concept of duality. Operators generally should not be thought of as being associated with one description or the other, nor should we think of an electron as switching between particle and wave. It is better just to think of it as having specific position in space or not, and whether or not it has specific position it is neither a wave nor a particle, but rather just a quantum mechanical entity.

The results of any experiments we perform with electrons are predicted very well by the mathematical system that has been developed for quantum mechanics, and some of that math has parallels with classical wave behavior and some with classical particle behavior and some with neither of them. Beyond that, we have very little understanding of what an electron actually is and how its behavior accords with our understanding of the physics of the macroscopic world.