Probabilities for system of two bosons

In summary, the question is asking about the probability of two indistinguishable bosons being in the |0>|0> state. The total state of the system can be |0>|0>, |1>|1>, or (1/sqrt2)(|0>|1>+|1>|0>). The most general state of the two particles is a|00\rangle+b|11\rangle+c|S\rangle with |a|^2+|b|^2+|c|^2=1. To have equal probabilities for each particle being in |0\rangle or |1\rangle, a=b=c and each would have to be 1/sqrt3, taking into account the 1/sqrt2 factor in
  • #1
ajl1989
8
0
If there are two indistinguishable bosons that can either be in the |0> or |1> state, what is the probability that both will be in the |0> state? (ie the system will be in the |0>|0> state)

I know there are only three possibilities for the total state of the system: |0>|0>, |1>|1>, and (1/sqrt2)(|0>|1>+|1>|0>), but are these states equally probable? (I'm assuming that a single boson has an equal probability of being in either |0> or |1>) Would the probability of being in |0>|0> just be 1/3? or is it more complicated than that?
 
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  • #2
Call the three states

[tex]|00\rangle,\;|11\rangle,\;\hbox{and}\;|S\rangle \equiv{\textstyle{1\over\sqrt2}}\bigl(|01\rangle+|10\rangle\bigr).[/tex]

Then the most general state of the two particles is

[tex]a|00\rangle+b|11\rangle+c|S\rangle[/tex]

with [itex]|a|^2+|b|^2+|c|^2=1[/itex].

What can you deduce about [itex]a[/itex], [itex]b[/itex], and [itex]c[/itex] from the requirement that each particle be equally likely to be in [itex]|0\rangle[/itex] or [itex]|1\rangle[/itex]?
 
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  • #3
Well, I guess if a=b=c, then they'd each have to be 1/sqrt3... unless we also have to take into account the 1/sqrt2 factor in front of |S>?
 

What is the concept of "Probabilities for system of two bosons"?

The concept of "Probabilities for system of two bosons" refers to the mathematical calculation and analysis of the likelihood of various outcomes in a system consisting of two bosons. Bosons are a type of elementary particle that follow Bose-Einstein statistics, which governs the behavior of particles in systems with integer spin.

Why is it important to study the probabilities for systems of two bosons?

Studying the probabilities for systems of two bosons allows us to understand and predict the behavior of these particles in various physical systems. This knowledge is crucial in fields such as quantum mechanics, condensed matter physics, and particle physics, where bosons play a significant role.

How are probabilities calculated for systems of two bosons?

The probability of finding two bosons in a particular state is calculated using the Bose-Einstein distribution, which takes into account the number of particles, the energy levels, and the temperature of the system. This distribution allows us to determine the likelihood of different configurations of two bosons in a given system.

Can probabilities be calculated for systems of more than two bosons?

Yes, probabilities can be calculated for systems of any number of bosons. However, the calculations become more complicated as the number of particles increases, and it may not be feasible to solve the equations analytically for large systems. In such cases, numerical methods are used to approximate the probabilities.

What is the significance of the probabilities obtained for systems of two bosons?

The probabilities obtained for systems of two bosons help us understand the behavior of these particles and make predictions about their behavior in various physical systems. This knowledge has practical applications in fields such as quantum computing, superfluidity, and superconductivity.

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