Wavefunctions for Indistinguishable and Distinguishable particles -

In summary, the conversation discusses the concept of wavefunctions for distinguishable and indistinguishable particles in a one-dimensional potential well. It also addresses the spin states of the particles and how they affect the wavefunction. The conversation also touches on the dependence of the wavefunction on time.
  • #1
captainjack2000
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0
Wavefunctions for Indistinguishable and Distinguishable particles - URGENT

Homework Statement


A one-dimensional potential well has a set of single-particle energy eigenstates Un(x) with energies En=E_o n^2 where n=1,2,3... Two particles are placed in the well with three possible sets of properties.
a)2 distinguishable spin 0 particles
b)2 identical spin 0 particles
c)2 identical spin 1/2 particles

Write down the spatial part of the two-particle wave functions at t=0 for the two lowest energy states of the two-particle system, and hence give the degeneracies of these energy states and explain how these two-particle wavefunctions depend on time


Homework Equations



The Attempt at a Solution


I am getting incredibly confused by two-particle wavefunctions and between the spatial and spin states...

a) If there are 2 indenticle spin 0 particles: The wavefunction must be symmetric then would the wavefunction just be
(|0>|1> + |1>|0>)/sqrt(2)

b) 2 identicle spin 1/2 particles: The wavefunction must be antisymmetric due to Pauli Exclusion Principle so
(|0>|1> - |1> |0>)/sqrt(2)

c) 2 indistinguishable particles (boson or fermion) would it just be a wavefunction with 6 dimensions
ie phi(r1,r2)

How would you then write the spin part of the wavefunction...and how do they depend on time...

Realise this might be all wrong but I have an exam coming up and would REALLY appreciate someone clarifying this!
 
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  • #2


Spin is one of those concepts that seems really hard to understand. The simple answer is that essentially all that is required is that opposite spins are orthogonal to each other in a particular basis.

So if you have spin up and spin down you might represent them as matrices like:

Spin up:[tex]\left[ \begin{array}{cc}1 \\ 0 \end{array} \right][/tex]Spin down:[tex]\left[ \begin{array}{cc}0 \\ 1 \end{array} \right][/tex]
 
  • #3


Does that mean that my wavefunctions look correct?

How would they differ for the two lowest energy states?
 

1. What is a wavefunction and how is it related to particles?

A wavefunction is a mathematical description of the quantum state of a particle or system of particles. It contains information about the probability of finding a particle in a certain position or with a certain momentum. In quantum mechanics, particles are described as both waves and particles, and the wavefunction is the mathematical representation of the particle's wave-like behavior.

2. What is the difference between distinguishable and indistinguishable particles?

Distinguishable particles are particles that can be easily differentiated from one another, such as particles with different masses or charges. Indistinguishable particles, on the other hand, are identical and cannot be differentiated based on any physical properties. This is a concept in quantum mechanics that is crucial in understanding the behavior of particles in systems.

3. How are the wavefunctions of distinguishable and indistinguishable particles different?

The wavefunctions for distinguishable particles follow classical probability rules, where the total wavefunction is the product of the individual wavefunctions. In contrast, the wavefunctions for indistinguishable particles must follow quantum mechanical principles, such as obeying the Pauli exclusion principle and being symmetric or anti-symmetric under particle exchange.

4. Can two particles have the same wavefunction?

No, according to the Pauli exclusion principle, no two identical particles can have the same quantum state. This means that their wavefunctions must be different, even if they are indistinguishable particles.

5. How do wavefunctions for indistinguishable particles play a role in explaining phenomena such as superconductivity and Bose-Einstein condensates?

In superconductivity and Bose-Einstein condensates, particles behave as if they are in the same quantum state, meaning they have the same wavefunction. This allows for the formation of a macroscopic quantum state, leading to unique properties such as zero resistance in superconductors and the ability for all particles to occupy the same quantum state in Bose-Einstein condensates.

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