Read Online Quantum Man: Richard Feynman's Life in Science by Lawrence M. Krauss - Free Book Online Page B
Authors: Lawrence M. Krauss
Tags: Science / Physics
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other out.
It is precisely this mathematics that is behind the behavior of electrons shot at a scintillating screen, as described in chapter 2. Here we find that an electron can actually interfere with itself because electrons have a nonzero probability of being in many different places at any one time.
Let’s first think about how to calculate probabilities in a sensible, classical world. Consider choosing to travel from town a to town c by taking some specific route through town b . Let the probability of choosing some route from a to b be given by P( ab ), and then the probability of choosing some specific route from b to c be P( bc ). Then, if we assume that what happens at b is completely independent of what happens at a and c , the probability of traveling from a to c along a specific route that goes through town b is simply given by the product of the two probabilities, P( abc ) = P( ab ) × P( bc ). For example, say there is a 50 percent chance of taking some route from a to b , and then a 50 percent chance of taking some route from b to c . Then if we were to send four cars out, two will make it to b on the chosen route, and of those two, one will take the next chosen route from b to c . Thus there is a 25 percent (.5 × .5) probability of taking the required route all the way from beginning to end.
Now, say we don’t care which particular point b is visited between a and c . Then the probability of traveling from a to c , given by P( ac ), is simply the sum of the probabilities P( abc ) of choosing to go through any point b between a to c .
The reason this makes sense is that classically if we are going from a to c , and b represents the totality of different towns we can cross through, say, halfway from both a and c , then we have to go through one of them during our journey (see figure).
(Since this picture is reminiscent of the earlier pictures of light rays, then we could say that if the example in question involved light rays going from a to c , then we could use the principle of least time to determine that the probability of going through one of the routes, that of least time, was 100 percent, and the probability of taking any other route was zero.)
The problem is that things don’t work this way in quantum mechanics. Because probabilities are determined by the squares of probability amplitudes to go from one place to another, the probability to go from a to c is not given by the sum of the probabilities to go from a to c via any definite intermediate point b . This is because in quantum mechanics it is the separate probability amplitudes for each part of the route that multiply and not the probabilities themselves. Thus, the probability amplitude to go from a to c through some definite point b is given by multiplying the probability amplitude to go from a to b times the probability amplitude to go from b to c .
If we don’t specify which point b to travel through, the probability amplitude to go from a to c is again given by the sum of the product of probability amplitudes to go from a to b and from b to c , for all possible b ’s. But this means that actual probability is now given as the square of the sum of these products. Since some terms in the sum can be negative, the crazy quantum behavior I discussed in chapter 2 for electrons hitting a screen can occur. Namely, if we don’t measure which of two points, say b and b ' , a particle traverses as it travels through one of two slits between a and c , then the probability of arriving at point c on the screen is determined by the sum of the squares of the probability amplitudes for the two different allowed paths. If we do measure which point, b or b ' , the particle traverses in between a and c , then the probability is simply the square of the probability amplitude for a single path. In the case of many electrons shot one at a time, the final pattern on the screen in the former case will be determined by adding the squares of the sum of
It is precisely this mathematics that is behind the behavior of electrons shot at a scintillating screen, as described in chapter 2. Here we find that an electron can actually interfere with itself because electrons have a nonzero probability of being in many different places at any one time.
Let’s first think about how to calculate probabilities in a sensible, classical world. Consider choosing to travel from town a to town c by taking some specific route through town b . Let the probability of choosing some route from a to b be given by P( ab ), and then the probability of choosing some specific route from b to c be P( bc ). Then, if we assume that what happens at b is completely independent of what happens at a and c , the probability of traveling from a to c along a specific route that goes through town b is simply given by the product of the two probabilities, P( abc ) = P( ab ) × P( bc ). For example, say there is a 50 percent chance of taking some route from a to b , and then a 50 percent chance of taking some route from b to c . Then if we were to send four cars out, two will make it to b on the chosen route, and of those two, one will take the next chosen route from b to c . Thus there is a 25 percent (.5 × .5) probability of taking the required route all the way from beginning to end.
Now, say we don’t care which particular point b is visited between a and c . Then the probability of traveling from a to c , given by P( ac ), is simply the sum of the probabilities P( abc ) of choosing to go through any point b between a to c .
The reason this makes sense is that classically if we are going from a to c , and b represents the totality of different towns we can cross through, say, halfway from both a and c , then we have to go through one of them during our journey (see figure).
(Since this picture is reminiscent of the earlier pictures of light rays, then we could say that if the example in question involved light rays going from a to c , then we could use the principle of least time to determine that the probability of going through one of the routes, that of least time, was 100 percent, and the probability of taking any other route was zero.)
The problem is that things don’t work this way in quantum mechanics. Because probabilities are determined by the squares of probability amplitudes to go from one place to another, the probability to go from a to c is not given by the sum of the probabilities to go from a to c via any definite intermediate point b . This is because in quantum mechanics it is the separate probability amplitudes for each part of the route that multiply and not the probabilities themselves. Thus, the probability amplitude to go from a to c through some definite point b is given by multiplying the probability amplitude to go from a to b times the probability amplitude to go from b to c .
If we don’t specify which point b to travel through, the probability amplitude to go from a to c is again given by the sum of the product of probability amplitudes to go from a to b and from b to c , for all possible b ’s. But this means that actual probability is now given as the square of the sum of these products. Since some terms in the sum can be negative, the crazy quantum behavior I discussed in chapter 2 for electrons hitting a screen can occur. Namely, if we don’t measure which of two points, say b and b ' , a particle traverses as it travels through one of two slits between a and c , then the probability of arriving at point c on the screen is determined by the sum of the squares of the probability amplitudes for the two different allowed paths. If we do measure which point, b or b ' , the particle traverses in between a and c , then the probability is simply the square of the probability amplitude for a single path. In the case of many electrons shot one at a time, the final pattern on the screen in the former case will be determined by adding the squares of the sum of
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