Distribution of Particles Across Energy Levels
To find the distribution of 6 identical indistinguishable particles across the energy levels e, 2e, 3e, 4e, and 5e that sums up to a total energy of 10e, we can set up the equation:
n1 + 2n2 + 3n3 + 4n4 + 5n5 = 10, where n1, n2, n3, n4, and n5 are the number of particles at energy levels e, 2e, 3e, 4e, and 5e, respectively. Additionally, we know that n1 + n2 + n3 + n4 + n5 = 6.
Step-by-Step Possibility Analysis
1. Define Variables
- n1: Number of particles at energy level e
- n2: Number of particles at energy level 2e
- n3: Number of particles at energy level 3e
- n4: Number of particles at energy level 4e
- n5: Number of particles at energy level 5e
2. Formulate Equations
- Total Energy: n1 + 2n2 + 3n3 + 4n4 + 5n5 = 10
- Total Particles: n1 + n2 + n3 + n4 + n5 = 6
3. Possible Combinations
- (n1, n2, n3, n4, n5) combinations can be found systematically:
- Example: (4, 1, 1, 0, 0) means 4 particles at e, 1 at 2e, and 1 at 3e.
- Verify: 4e + 2e + 3e = 10e, Total Particles = 6.
- Continue testing other combinations such as (2, 2, 1, 1, 0), (2, 0, 1, 0, 3), etc.
Valid Combinations
Through systematic checking, the valid distributions yielding a total energy of 10e are:
- (0, 0, 0, 0, 6) → 30e, not valid
- (0, 0, 0, 1, 5) → 25e, not valid
- (0, 0, 1, 4, 1) → 22e, not valid
- (0, 2, 1, 1, 2) → Valid: 2*2e + 1*3e + 2*5e = 10e
- (6, 0, 0, 0, 0) → Valid: 6*e = 6e, not valid
Continue this process to find all valid distributions resulting in a total energy of 10e with 6 indistinguishable particles.