An operator is said to be an observable if all eigen kets form basis in the whole state space.

In quantum mechanics, an observable refers to a physical quantity or property of a system that can be measured. Observables are represented by mathematical operators, and when applied to a quantum state, they provide information about the possible outcomes of measurements related to that property.

Here's a closer look at what observables are in quantum mechanics:

Physical Properties: Observables represent physical properties of a quantum system, such as position, momentum, energy, angular momentum, spin, and more. Each observable corresponds to a particular aspect of the system that we might want to measure.
Mathematical Operators: In quantum mechanics, observables are represented by mathematical operators. These operators act on the wave function (or state vector) of the system, transforming it into another wave function that represents the state after the measurement.
Measurement Outcomes: When you measure an observable, you get a specific value as the outcome. However, quantum mechanics introduces a probabilistic nature to measurements. The outcome you get is based on the probabilities associated with different possible values for the observable.
Spectrum: Each observable has a range of possible values called its spectrum. The spectrum can be continuous (like the position of a particle) or discrete (like the possible energy levels of an atom).
Expectation Value: The expectation value of an observable is the average value you would expect to measure if you performed many measurements on identical systems prepared in the same quantum state. It's a crucial concept because it helps predict the most likely outcome of measurements.
Eigenvalues and Eigenvectors: Observables have special states called eigenstates (or eigenvectors) associated with them. When you measure an observable, the system is most likely to collapse into one of these eigenstates, and the corresponding eigenvalue is the value you will measure.
Commutation Relations: Observables may not commute with each other, meaning their order of measurement can affect the outcomes. This is described by commutation relations, and it's a fundamental aspect of quantum mechanics.
Let
be orthogonal eigen ket of A corresponding to eigen value λ1. These eigen kets constitute orthonormal basis in 𝜖_λ1.
Let
be orthogonal eigen ket of A corresponding to eigen value λ2. These eigen kets constitute orthonormal basis in 𝜖_λ2.
be orthogonal eigen ket of A corresponding to eigen value λn. These eigen kets constitute orthonormal basis in 𝜖_λn.
The condition for the formation of basis by all these eigen vectors in whole state space is
This is the condition for A to be an observable. We show that the projection operator P_𝜓 is an observable. First we show the eigen values of P_𝜓 are 1 and 0.
Let P_𝜓 l f> = λl f>
If λ is not zero l 𝜓> and l f> are collinear (in the same direction). <𝜓l f>= 1 then λ = 1 non degenerate.
We show that if l 𝜙 > is any ket of the state space, then P_𝜓 l 𝜙 > is eigen ket of P_𝜓 corresponding to eigen value λ = 1 and (I - P_𝜓) l 𝜙 > is eigen ket of P_𝜓 corresponding to eigen value λ = 0.
P_𝜓 (P_𝜓) l 𝜙 > = P_𝜓² l 𝜙 > = P_𝜓) l 𝜙 >
When we compare this relation with P_𝜓 (P_𝜓 l 𝜙 >) = λ(P_𝜓 l 𝜙 >)
λ = 1, so P_𝜓 l 𝜙 > is eigen ket of P_𝜓 corresponding to eigen value λ = 1
P_𝜓 ( I - P_𝜓 )l 𝜙 > = P_𝜓( l 𝜙 > - P_𝜓 l 𝜙 >) = P_𝜓 l 𝜙 > - P_𝜓² l 𝜙 > = P_𝜓 l 𝜙 > - P_𝜓 l 𝜙 > = 0 ( I - P_𝜓 )l 𝜙 >
then λ = 0.
( I - P_𝜓 )l 𝜙 > is eigen ket of P_𝜓 corresponding to eigen value λ = 0. We can expand any ket l 𝜓 > as
l 𝜙 > = P_𝜓 l 𝜙 > + ( I - P_𝜓 )l 𝜙 >
l 𝜙 > = C1 P_𝜓 l 𝜙 > + C2 ( I - P_𝜓 )l 𝜙 >
Thus eigen kets of P_𝜓 form basis. So P_𝜓 is an observable.
In summary, observables in quantum mechanics are the quantities or properties we want to measure in a quantum system. They're represented by mathematical operators, and their measurement outcomes are
probabilistic in nature. Observables provide a way to understand and predict the behaviour of quantum systems by allowing us to extract information about their physical properties through measurements.

This note is a part of the Physics Repository.

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