Measuring the polarisation of light

Question

I've been thinking about the standard and easy "Three-polarising-filters experiment" as a demonstration of quantum phenomenon:

If light is passed through two polarising filter and the filters have relative angle of 90° then no light will pass. If a third polarising filter is inserted between the initial two, with some relative angle, then light passes.

and I was wondering whether:

The insertion of polarising filters could be thought of as "asking the light" what is its component in the specific basis (specified by the filter), with the caveat that after each "question" (i.e. insertion of polarising filter) the light polarisation effectively changes basis and only the component¹ that coincides with the allowed (fast) axis of the filter continues through?

Could this be generalised to every measurement, i.e. the measuring device is defining a basis (in the sense of vector basis) and is not only measuring a component of the object of interest in the specified basis, but also performing a change of basis on it, which remains after the measurement is done?

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^{1.If we think that in each basis the light polarisation is consisted of two components which are at right angles (orthogonal) to each other.}

Answer 1

Yes, in quantum mechanics observables are represented by Hermitian operators, which have real eigenvalues and an orthonormal basis of eigenstates. An ideal measurement of an observable $X$ selects an eigenstate randomly according to the probability distribution specified by the system's state, returns the eigenstate's eigenvalue as the measurement, and then projects the system's state onto that eigenstate. This last step is often called " the collapse of the wavefunction."

Answer 2

If you are given a Hermitian operator $\Lambda$ acting in a Hilbert space, this operator will have eigenstates $\vert\lambda_i\rangle$ with eigenvalues $\lambda_i$. Ignoring multiplicities of $\lambda_i$ for simplicity, an ideal measurement for the eigenvalue $\lambda_i$ is then represented by the projector $$ \Pi_i =\vert\lambda_i\rangle\langle \lambda_i\vert\, . $$ Note that $\Pi_i$ has $\vert \lambda_i\rangle$ as eigenstate with eigenvalue $1$. All the other eigenstates of $\Lambda$ are eigenstate of $\Pi_i$ with eigenvalue $0$, i.e. $$ \Pi_i\vert\lambda_j\rangle =\delta_{ij} $$ With this, the formalism states that, if you measure $\Lambda$ and obtain the value $\lambda_i$, the initial state $\vert\psi\rangle$ becomes $$ \vert\psi\rangle \to \Pi_i\vert\psi\rangle = \vert\lambda_i\rangle \langle \lambda_i\vert\psi\rangle \tag{1} $$ i.e. the initial state projects to the eigenstate with eigenvalue $\lambda_i$. The quantity $\vert\langle \lambda_i\vert\psi\rangle\vert^2$ is the probability of getting the outcome $\lambda_i$ when measuring $\Lambda$.

The state $\vert\lambda_i\rangle\langle \lambda_i\vert\psi\rangle$ is not normalized and must be normalized “by hand”: $\vert\lambda_i\rangle$ becomes the “new” normalized initial state immediately after the measurement.

In this sense, the measurement is not quite like asking “what is the component” in a specific basis, but rather “is there a component in a specific basis”. Whereas the answer to the first question is a complex number, the answer to the second is binary: either “yes” (if $\langle \lambda_i\vert \psi\rangle \ne 0$) or “no” (if $\langle \lambda_i\vert\psi\rangle =0$).

In the specific case of your example, you have $\vert \uparrow\rangle$ as initial state and the measurement operator $\Pi_{\nearrow}=\vert\nearrow\rangle\langle \nearrow\vert$ so that $$ \Pi_{\nearrow}\vert \uparrow\rangle = \vert\nearrow\rangle \langle \nearrow\vert \uparrow\rangle $$ can be interpreted as asking if your initial state has some “$\vert \nearrow\rangle$ light in it”.

Of course, as you have correctly guessed $$ \Pi_\rightarrow\vert \uparrow\rangle =0 $$ but $$ \Pi_\rightarrow\Pi_\nearrow \vert\uparrow\rangle \ne 0 $$ i.e. inserting a filter with an intermediate orientation changes the result because now $\Pi_\rightarrow$ is like asking if there is some $\vert\rightarrow\rangle$ in $\vert \nearrow\rangle$, not in $\vert\uparrow\rangle$.