Eigenvalues and Measurement

We now arrive at one of the most important concepts of quantum mechanics: the possible outcomes of a measurement of a quantity corresponding to $A$ are only the eigenvalues of $A$. After the measurement the system is in an eigenstate of $A$ with a predictable probablity depending on $A$ and its state just before the measurement.

These two statements belong to the Kopenhagen interpretation of quantum mechanics and are widely accepted and experimentally confirmed by now. They belong to the axioms of quantum mechanics and can be motivated as follows:

Consider a quantum mechanical system in a normalized state $\vert\psi\rangle$. We wish to perform a measurement of a quantity (for example the energy) that is represented by a hermitian operator $A$ (for example the Hamiltonian $\hat{H}$).

CASE 1: Assume that $\vert\psi\rangle=\vert\phi\rangle$ is an eigenstate of $A$, $A\vert\phi\rangle=\lambda\vert\phi\rangle$ with eigenvalue $\lambda$. Repeating this measurement at many systems that are prepared in the same way, or at the same system that is always prepared in the same state $\vert\phi\rangle$, the expectation value of $A$ is $\langle A \rangle = \lambda$.

CASE 2: Assume that $\vert\psi\rangle$ is not an eigenstate of $A$. After a measurement with outcome $a$ assume the system is in another state $\vert\phi\rangle$. Assume immediately after the first measurement, a second measurement with outcome $b$ is performed. We now assume that this second measurement should give the same outcome as the first measurement, i.e. $a=b$. This thought experiment is repeated many times at identically prepared systems. Always $b=a$ should come out such that the square deviation of $A$ for the state $\vert\phi\rangle$ is zero:

$$\langle [A- \langle A \rangle ]^2 \rangle = \frac{\langle \phi [... ...i \rangle }{\langle \phi \vert\phi\rangle}=0 \leadsto (A-a)\vert\phi \rangle=0.$$

This tells us that after the first measurement, the system is in an eigenstate $\vert\phi\rangle$ of $A$, and the outcome of this measurement is an eigenvalue $a$ of $A$. The second measurement then is as in case 1 and yields $a$ with the system remaining in the eigenstate $\vert\phi\rangle$.

Axiom 2c: The possible outcomes of measurements of an observable corresponding to the hermitian linear operators $A$ are the eigenvalues of $A$. Immediately after the measurement, the quantum system is in the eigenstate of $A$ corresponding to the eigenvalue that is measured.

This axiom is the most radical break with classical physics: it postulates an abrupt collapse of the wave function (`reduction of the wave packet') into one of the eigenstates of $A$, if a measurement is performed. Before the measurement is actually done, one can not predict its outcome, that is which eigenvalue is measured. Only probabilities for the possible outcomes can be predicted:

Axiom 2d: Let $A$ have a complete system of eigenvectors $\{\vert\phi_n\rangle\}$ with eigenvalues $a_n$. The normalized state $\vert\psi\rangle$ before the measurement of $A$ can be expanded into

$$\vert\psi\rangle=\sum_{n=0}^{\infty}c_n\vert\phi_n\rangle,\quad c_n = \langle\phi_n\vert\psi\rangle \in C.$$ (195)

Then, the expectation value of $A$ in $\vert\psi\rangle$ is

$$\langle\vert \psi \vert A\vert\psi\rangle =\sum_{n=0}^{\infty}a_n \vert c_n\vert^2=:\sum_{n=0}^{\infty}a_n p_n,$$ (196)

and the probability $p_n$ to find the system in the eigenstate $\vert\phi_n\rangle$ after the measurement is given by the amplitude square $p_n = \vert c_n\vert^2=\vert\langle\phi_n\vert\psi\rangle\vert^2$.