Read Online Quantum Man: Richard Feynman's Life in Science by Lawrence M. Krauss - Free Book Online Page A
Authors: Lawrence M. Krauss
Tags: Science / Physics
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how the classical motion of a baseball evolves in time, or Maxwell’s equations, which tell us how electromagnetic waves evolve in time. The difference is that in quantum mechanics the quantity that evolves in time in a deterministic manner is not directly observable, but rather is a set of probabilities for making certain observations, in this case for determining the particle to be at a certain place at a certain time.
This is strange enough, but it further turns out that the wave function itself does not directly describe the probability of finding a particle at a given place at some time. Instead it is the square of the wave function that gives the probabilities. This one fact is responsible for all of the strangeness of the quantum mechanical world because it explains why particles can behave precisely as waves, as I will describe now.
First, note that the probabilities of things we measure must be positive (we would never say that there is a probability of minus 1 percent of finding something) and the square of a quantity is also always positive, so quantum mechanics predicts positive probabilities—which is a good thing. But it also implies that the wave function itself can be either positive or negative, since, say, −½ and +½ both yield the same number (+¼) when squared.
If it were the wave function that described the probability of finding some particle at some location x , then if I had two identical particles, the probability of finding either particle at location x would be the sum of the two individual (and each necessarily positive) wave functions. However, because the square of the wave function is what determines the probability of finding particles, and because the square of the sum of two numbers is not equal to the sum of the individual squares, things can get much more interesting in quantum mechanics.
Let’s say the value of the wave function that corresponds to finding particle A at position x is P1, and the value of the wave function that corresponds to finding particle B at position x is P2, then quantum mechanics tells us that the probability of finding either particle A or B at position x is now (P1 + P2) 2 . Let’s say P1 = ½ and P2 = −½. Then if we only had one particle, say particle A , the probability of finding it at position x would be (½) 2 = ¼. Similarly the probability of finding particle B at position x would be (−½) 2 = ¼. However, if there are two particles, the probability of finding either particle at position x is ((½) + (−½)) 2 = 0.
This phenomenon, which on the surface seems ridiculous, is in fact familiar for waves, say, sound waves. Such waves can interfere with each other so that, for example, waves on a string can interfere and produce locations on the string, called nodes , that do not move at all. Similarly, if sound waves are coming from two different speakers in a room, we might find, if we were to walk around the room, certain locations where the waves cancel each other out, or, as physicists say, negatively interfere with each other. (Acoustic experts design concert halls so that hopefully there are no such “dead spots.”)
What quantum mechanics, with probabilities being determined by the square of the wave function, tells us is that particles too can interfere with each other, so that if there are two particles in a box, the probability of finding either of them at a given location can end up being less than the probability of finding one where only a single particle is in the box.
When waves interfere, it is the height, or amplitude , of the resulting wave that is affected by this interference, and the amplitude can be positive or negative depending on whether one is at a peak or a trough in the wave. So another name for the wave function of a particle is its probability amplitude , which can be positive or negative.
And just as for regular amplitudes for sound waves, separate probability amplitudes for different particles can cancel each
This is strange enough, but it further turns out that the wave function itself does not directly describe the probability of finding a particle at a given place at some time. Instead it is the square of the wave function that gives the probabilities. This one fact is responsible for all of the strangeness of the quantum mechanical world because it explains why particles can behave precisely as waves, as I will describe now.
First, note that the probabilities of things we measure must be positive (we would never say that there is a probability of minus 1 percent of finding something) and the square of a quantity is also always positive, so quantum mechanics predicts positive probabilities—which is a good thing. But it also implies that the wave function itself can be either positive or negative, since, say, −½ and +½ both yield the same number (+¼) when squared.
If it were the wave function that described the probability of finding some particle at some location x , then if I had two identical particles, the probability of finding either particle at location x would be the sum of the two individual (and each necessarily positive) wave functions. However, because the square of the wave function is what determines the probability of finding particles, and because the square of the sum of two numbers is not equal to the sum of the individual squares, things can get much more interesting in quantum mechanics.
Let’s say the value of the wave function that corresponds to finding particle A at position x is P1, and the value of the wave function that corresponds to finding particle B at position x is P2, then quantum mechanics tells us that the probability of finding either particle A or B at position x is now (P1 + P2) 2 . Let’s say P1 = ½ and P2 = −½. Then if we only had one particle, say particle A , the probability of finding it at position x would be (½) 2 = ¼. Similarly the probability of finding particle B at position x would be (−½) 2 = ¼. However, if there are two particles, the probability of finding either particle at position x is ((½) + (−½)) 2 = 0.
This phenomenon, which on the surface seems ridiculous, is in fact familiar for waves, say, sound waves. Such waves can interfere with each other so that, for example, waves on a string can interfere and produce locations on the string, called nodes , that do not move at all. Similarly, if sound waves are coming from two different speakers in a room, we might find, if we were to walk around the room, certain locations where the waves cancel each other out, or, as physicists say, negatively interfere with each other. (Acoustic experts design concert halls so that hopefully there are no such “dead spots.”)
What quantum mechanics, with probabilities being determined by the square of the wave function, tells us is that particles too can interfere with each other, so that if there are two particles in a box, the probability of finding either of them at a given location can end up being less than the probability of finding one where only a single particle is in the box.
When waves interfere, it is the height, or amplitude , of the resulting wave that is affected by this interference, and the amplitude can be positive or negative depending on whether one is at a peak or a trough in the wave. So another name for the wave function of a particle is its probability amplitude , which can be positive or negative.
And just as for regular amplitudes for sound waves, separate probability amplitudes for different particles can cancel each
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