The beam of white light is refracted by the prism, but the different wavelengths in the light are refracted by different angles. That means when the light falls on the screen the different wavelengths are spread out over a region of the screen. This is what is meant by dispersion.
The dispersion is linear if the different wavelengths of light are spread out evenly across the screen. So the wavelength at the point I've shown as $d$ would be given by:
$$ \lambda_d = \lambda_{red} - \frac{d \left(\lambda_{red} - \lambda_{blue}\right)}{L} $$
In general the dispersion won't be linear, but for glass across the range of red to blue light it's not a bad approximation. The reason this matters is when you try to calculate the coherence length from the size of the slit. The coherence length is given by:
$$ L = \frac{2\space ln(2)}{\pi n} \frac{\lambda^2}{\Delta \lambda} $$
Because the slit has a finite width the image is casts on the screen has a finite width and therefore covers a range of frequencies. Going back to my diagram above, suppose the image of the slit on the screen has a width of $w$, then if the dispersion can be assumed linear the range of frequencies is simply:
$$ \Delta \lambda = \frac{w}{L} \left(\lambda_{red} - \lambda_{blue} \right) $$
Put this value for $\Delta \lambda$ back into the equation for the coherence length to get the effect of the slit width on coherence length.
To start with the double slit experiment gives interference even when the beam is composed by one photon at a time. The spot on the screen a photon/particle the statistical accumulation the interference seen as expected classically too.
The joint comes because the photon as a quantum mechanical entity has a wavefunction that is the solutions of Maxwell's equation, treated as operators operating on the wave function. The E=h*nu identity the photon carries is the same as the frequency of the classical solution of Maxwell's equation and that , together with the phase attached to the wavefunction allow the continuity of classical down to quantum single photon level For a formal treatment how classical emerges from an ensemble of photons have a look at this blog entry.
There exists a fundamental difference between classical and quantum equations and their solutions, but also a continuity at the interface.
Edit after question edit
How is the theory of partial coherent light related to quantum-mechanics?
This needs somebody familiar with the formalism of both, but I believe the connection should follow the method in the link, how classical electromagnetic beams emerge from an ensemble of photons.
]>... the amplitude of a wave function ... But is this really a fundamental difference, or just a difference in the common practices of the respective theories?
the square of the wavefunction is the connection with predictions and experiments in quantum mechanics, it is the probabilistic nature that makes the difference with the classical framework, as far as I know.
How much of the strange phenomena of quantum-mechanics can be explained by the theory of partial coherent light alone, without any reference to particles or measurement processes?
Phenomenon is " an observable" , observing something implies a measurement process, measurement implies interaction, picking up a point that will contribute to the quantum mechanical probability distribution (or building up the distribution itself by continuous observations) so there is an inherent contradiction in this part of the question.
Best Answer
The coherence length is just the coherence time multiplied by the propagation speed.
To understand the coherence time, say you have a wave described, in complex notation, by $$ E(t) = A(t) e^{i \omega t} $$ where $A(t)$ is a slowly varying complex amplitude. You make this wave interfere with a delayed version of itself and collect the intensity $$ |E(t) + E(t-\tau)|^2 = |E(t)|^2 + |E(t-\tau)|^2 + 2\Re\big(E(t)E^*(t-\tau)\big). $$ where $\Re$ means real part and $^*$ means complex conjugate. The interference term is $$ 2\Re\big(E(t)E^*(t-\tau)\big) = 2\Re\big(A(t)A^*(t-\tau)e^{i \omega \tau}\big) $$ If $A(t)$ is constant, or roughly constant within a time interval $\tau$, then this becomes $$ 2|A(t)|^2 \cos(\omega \tau) $$ which is the interference pattern. On the other hand, if $A(t)$ fluctuates sufficiently fast, and $\tau$ is larger than its correlation time, then $A(t)A^*(t-\tau)$ averages to zero and there is no interference. Thus, the coherence time can be simply seen as the correlation time of the complex amplitude $A(t)$.
Now, I'm not sure there is a very quantitative definition of the correlation time. You could define it as the delay where the autocorrelation function drops below some arbitrary threshold. This is equivalent to setting a threshold on the visibility of the interference pattern. The relationship with the shape of the spectral line should also be apparent: the squared modulus of the Fourier transform of $A(t)$ is the shape of the line (the spectrum of the wave shifted by $-\omega$). It is also the Fourier transform of the autocorrelation function of $A(t)$. Thus, when the line is wide, the autocorrelation function is narrow,and the coherence time is short.