Probability problem from Griffiths QM

In summary, the problem at hand is to find the probability that a needle of length l dropped at random onto a sheet of paper ruled with parallel lines a distance l apart will cross a line. The solution involves finding the probability of the needle hitting a line as a function of the position of its center relative to the lines, integrating it from 0 to l, and dividing by l. This results in a value that is inversely proportional to pi due to the integration over the orientation of the needle.
  • #1
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Howdy everybody. I've browsed this forum many a time but this is my first time posting. Anyways. I'm taking QM in the fall for the second time (the sole reason i didn't graduate this semester), and so I'm going through the griffiths book right now so i can ace this class next time around. I'm sort stuck on this problem (1.13). I'm researching for the university right now and so i could go talk to the prof, but (a) i don't want to look like to much of a kiss-ass by letting him know that i am already studying for his class and (b) I'm semi-terrified of this professor. so voila, here i am. any help would great.

"a needle of length l is dropped at random onto a sheet of paper ruled with parallel lines a distance l apart. what is the probability that the needle will cross a line?"

in the previous problem. i solved for the expectation values of theta, and its <x> projection. I'm thinking the solution will be a combination of the probabilities for theta, and also for where the needle falls on the y-axis (i.e. right on a line or somewhere between two lines) but i can't quite get a handle on the problem. any ideas?
 
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  • #2
If the center of the needle lands on a line the probability it will hit it is 1. If it lands half way between the lines the probability is 0. So figure the probability of hitting the lines as a function of the position of it's center relative to the lines. Then integrate that from 0 to l and divide by l.
 
  • #3
This is a classic problem, I think you should get a value that is inversely proportional to [tex]\pi[/tex] if you do it right.

I think once you integrate over the center-of-mass business that Dick mentioned you also integrate over it's orientation measured by an angle [tex]\theta[/tex] as you mentioned, and there are only two orientations where the needle still hits the lines (only looking at the case where the center-of-mass is midway between two lines), but dividing by the integral over all variables, including the angle one will get you that silly pi at the end.
 

1. What is a probability problem in Griffiths QM?

A probability problem in Griffiths QM is a problem that involves calculating the likelihood of a particular outcome or event occurring in a quantum mechanical system. This can involve finding the probability of a particle being in a certain state or location, or the probability of a measurement yielding a specific result.

2. How do you approach solving a probability problem in Griffiths QM?

The first step in solving a probability problem in Griffiths QM is to identify the system and the relevant variables, such as the state of the system, the observable being measured, and any known information or constraints. Then, you can use mathematical tools and concepts from quantum mechanics, such as the wave function and operators, to calculate the probability of the desired outcome.

3. What are some common techniques used in solving probability problems in Griffiths QM?

Some common techniques used in solving probability problems in Griffiths QM include using the Schrödinger equation to find the wave function, applying the Born rule to determine the probability amplitude, and using the normalization condition to ensure the total probability adds up to 1. Other techniques, such as using the uncertainty principle or the time-dependent Schrödinger equation, may also be applicable depending on the specific problem.

4. How does quantum mechanics differ from classical mechanics in terms of probability?

In classical mechanics, probabilities are seen as arising from our lack of knowledge about the system, and are described using statistical mechanics. In quantum mechanics, probabilities are fundamental and arise from the probabilistic nature of quantum systems. The wave function and the Born rule are used to calculate probabilities in quantum mechanics, while classical mechanics uses deterministic equations of motion.

5. Can you provide an example of a probability problem in Griffiths QM?

One example of a probability problem in Griffiths QM is the calculation of the probability of a particle initially in a superposition state being measured to have a particular spin value along a certain direction. This involves using the spin operator and the normalization condition to determine the probability amplitude, and then applying the Born rule to calculate the probability of obtaining the desired measurement outcome.

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