I am able to solve Schrodinger's equation for particles in vacuum and know many books that describe many variations of this problem and cover al technical details of the differential equation solving. However, it is unclear to me how quantum effects relate to the macroscopic world. How does a certain wave function relate to a number on the display of a (macroscopic) measuring device? Does anyone know a good book that covers this subject? I have a background in mathematics, so a bit technical is not a problem, but I am mostly interested in the general picture.
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Most good quantum mechanics books will include the meaning of the wavefunction in the introductory chapters. However it might be easier if I just briefly explained it here.
The Schrodinger equation allows you to solve for a wave function $\psi(x,t)$. This wave function is the probability amplitude. The probability of finding a particle between 2 points $a$ and $b$ is given by:
$$ P=\int^b_a\psi^*\psi \ dx $$
You cannot measure the wave function directly. However you can measure the position of particles. With one particle, you cannot retrieve the wavefunction. However if you keep measuring the position of particles and plot 'amount of particles' against position, it will be the same shape as the modulus square of the wavefunction.
Here is a good example of what I mean using the double slit experiment. Each electron has the exact same wave function however it is detected at a random spot. When many measurements have been made you can see the alternating nature of the wave function.
If you want more detail, the idea that the wavefunction is a probability is the 'Born rule'.