Consider two different systems each with three identical non-interacting particles. Both have single particle states with energies 𝜖₀, 3𝜖₀, and 5𝜖₀. One system is populated by spin... Consider two different systems each with three identical non-interacting particles. Both have single particle states with energies 𝜖₀, 3𝜖₀, and 5𝜖₀. One system is populated by spin-1/2 fermions and the other by bosons. What is the value of E_f - E_b where E_f and E_b are the ground state energies of the fermionic and bosonic systems respectively?
Understand the Problem
The question is asking to calculate the difference in ground state energies of two systems: one with fermions and another with bosons, based on given energy states. The high-level approach involves using the provided energy states to find the total energy for both types of particles and then determining the difference between the two.
Answer
The value of \( E_F - E_B \) is \( 7\epsilon_0 \).
Answer for screen readers
The value of ( E_F - E_B ) is ( 7\epsilon_0 ).
Steps to Solve
- Calculate the Energy of Fermions
For fermions, each particle respects the Pauli exclusion principle, meaning no two particles can occupy the same state. The available states are ( \epsilon_0 ), ( \epsilon_3 ), and ( \epsilon_6 ).
The ground state energy for 3 fermions is:
[ E_F = \epsilon_0 + \epsilon_3 + \epsilon_6 = \epsilon_0 + 3\epsilon_0 + 6\epsilon_0 = 10\epsilon_0 ]
- Calculate the Energy of Bosons
Bosons do not follow the exclusion principle, and multiple particles can occupy the same state. The lowest energy configuration with 3 bosons is:
[ E_B = 3 \epsilon_0 = 3\epsilon_0 ]
- Find the Difference in Energies
Now, we need to find the difference between the ground state energies of fermions and bosons:
[ E_F - E_B = 10\epsilon_0 - 3\epsilon_0 = 7\epsilon_0 ]
- Correctly Identify and Confirm the Result
From the previous calculation:
[ E_F - E_B = 7\epsilon_0 ]
Ensure to revisit the possible answer choices to confirm if any correction is needed, as it appears none directly match.
The value of ( E_F - E_B ) is ( 7\epsilon_0 ).
More Information
In this context, the difference in energies demonstrates the fundamental distinctions between fermionic and bosonic particles in quantum mechanics. Fermions, adhering to the Pauli exclusion principle, result in higher ground state energy configurations compared to bosons.
Tips
One common mistake is miscalculating the energies for fermions by not properly considering the restrictions imposed by the Pauli exclusion principle. To avoid this, ensure to account for each energy state occupied by separate fermions.
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