So I read that by applying an operator to the wavefunction (aka. measuring stuff),
It is one of the postulates of quantum mechanics:
With every physical observable q there is associated an operator Q, which when operating upon the wavefunction associated with a definite value of that observable will yield that value times the wavefunction.
Associated with any particle moving in a conservative field of force is a wave function which determines everything that can be known about the system.
it is as if the wavefunction collapses onto one defined state which is an eigenstate. Which specific state it actually is seems to be "chosen" completely at random (i.e. "God playing dice").
What waves in the wavefunction? In postulate number 5
. For a system described by a given wavefunction, the expectation value of any property q can be found by performing the expectation value integral with respect to that wavefunction.
To relate a quantum mechanical calculation to something you can observe in the laboratory, the "expectation value" of the measurable parameter is calculated. For the position x, the expectation value is defined as
where x is the x position operator.
This integral can be interpreted as the average value of x that we would expect to obtain from a large number of measurements. Alternatively it could be viewed as the average value of position for a large number of particles which are described by the same wavefunction. For example, the expectation value of the radius of the electron in the ground state of the hydrogen atom is the average value you expect to obtain from making the measurement for a large number of hydrogen atoms. etc in the link
etc in the link.
What waves is the probability of getting a specific measurement. That is why measuring the electron scattering off two slits gives an interference pattern. The individual measurement is a point on the screen, the distribution shows the wave solution, i.e. te probability of finding the electron at that (x,y) on the screen.
Measuring stuff, one individual measurement is the throw of the dice, has a probability of displaying the value. The dice has a probability of 1/6 of throwing a five, the measurement (throw) of a single electron in the double slit experiment gives a throw from the probability distribution.
Now the expectation value of the energy operator, will in a similar manner give a single value for the energy, which will be one instance in the probability distribution for that particular potential problem. It is a fact that energy levels are concentrated in value around a central value because there exists a width to the energy lines.
At the very same time, we are told that the eigenstate in question is not random at all for the Hamiltonian (lower energies are favoured and at 0K, we are in the ground state).
Systems relax to the lowest energy state is a principle through all physics frameworks, in quantum mechanics it appears as
The ground state of a quantum mechanical system is its lowest-energy state; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. In the quantum field theory, the ground state is usually called the vacuum state or the vacuum.
The solutions of the Hamiltonian are mathematical. A particular physical system is modeled by it. A single hydrogen atom will be in the ground state of the solution. To measure this, one has to scatter with a photon with the energy ( within the width) of the difference between the lowest state and the next one, excite the atom, i.e. a different wave function represents it, and verify the measurement by the relaxation and emission of a photon.
By the way, collapse is an unfortunate term to use for a probability distribution. A specific measurement described by a probability distribution classical or quantum mechanical does not collapse anything. It gives a probability of one for that single measurement to have happened. The accumulation of measurements will give the probability distribution, i.e. the width of the ground state in the hydrogen example.