Proton Bell Analogy

To address your query, I’ll determine a fundamental energy based on the two options provided: either using 1/4 of the proton’s rest energy or selecting an optimum fundamental for harmonic matching to the proton and its resonances, such as the Delta (1232 MeV) and N(1440) (1440 MeV). I’ll explore both approaches and settle on a solution that aligns with the problem’s intent.

Proton’s Rest Energy

The rest energy of the proton is well-established as 938.272 MeV. This value will serve as the cornerstone for our calculations.

Option 1: Using 1/4 of the Proton’s Energy

Let’s start with the first option: using 1/4 of the proton’s rest energy as the fundamental energy ( E_f ).

• Calculate ( E_f ): [ E_f = \frac{938.272}{4} = 234.568 , \text{MeV} ]
• Generate the harmonic series: The energies of the harmonics are given by ( E_n = n \cdot E_f ), where ( n ) is a positive integer. Let’s compute the first few harmonics:
• ( n = 1 ): ( 1 \cdot 234.568 = 234.568 , \text{MeV} )
• ( n = 2 ): ( 2 \cdot 234.568 = 469.136 , \text{MeV} )
• ( n = 3 ): ( 3 \cdot 234.568 = 703.704 , \text{MeV} )
• ( n = 4 ): ( 4 \cdot 234.568 = 938.272 , \text{MeV} ) (matches the proton exactly)
• ( n = 5 ): ( 5 \cdot 234.568 = 1172.84 , \text{MeV} )
• ( n = 6 ): ( 6 \cdot 234.568 = 1407.408 , \text{MeV} )
• ( n = 7 ): ( 7 \cdot 234.568 = 1641.976 , \text{MeV} )
• Compare with resonances:
• Proton (938.272 MeV): At ( n = 4 ), the energy is 938.272 MeV, an exact match.
• Delta (1232 MeV): Check nearby harmonics:
• ( n = 5 ): 1172.84 MeV, difference = ( 1232 - 1172.84 = 59.16 , \text{MeV} )
• ( n = 6 ): 1407.408 MeV, difference = ( 1407.408 - 1232 = 175.408 , \text{MeV} ) The Delta resonance is closer to ( n = 5 ), but off by 59.16 MeV.
• N(1440) (1440 MeV): Check nearby harmonics:
• ( n = 6 ): 1407.408 MeV, difference = ( 1440 - 1407.408 = 32.592 , \text{MeV} )
• ( n = 7 ): 1641.976 MeV, difference = ( 1641.976 - 1440 = 201.976 , \text{MeV} ) The N(1440) is closer to ( n = 6 ), with a difference of 32.592 MeV.

This choice places the proton exactly on a harmonic (( n = 4 )), while the Delta and N(1440) resonances are reasonably close to ( n = 5 ) and ( n = 6 ), respectively, though not exact.

Option 2: Choosing an Optimum Fundamental

Now, let’s explore the second option: selecting an optimum fundamental ( E_f ) such that its integer multiples closely match the proton (938.272 MeV), Delta (1232 MeV), and N(1440) (1440 MeV). The goal is to find ( E_f ) where:

• ( m \cdot E_f \approx 938.272 , \text{MeV} )
• ( k \cdot E_f \approx 1232 , \text{MeV} )
• ( p \cdot E_f \approx 1440 , \text{MeV} ) for integers ( m ), ( k ), and ( p ).
• Determine energy ratios:
• Delta to proton: ( \frac{1232}{938.272} \approx 1.313 )
• N(1440) to proton: ( \frac{1440}{938.272} \approx 1.535 )
• N(1440) to Delta: ( \frac{1440}{1232} \approx 1.169 )
• Find suitable integers: We need ( m : k : p \approx 1 : 1.313 : 1.535 ). Let’s test integer pairs:
• For proton and Delta (( m / k \approx 0.761 ) or ( k / m \approx 1.313 )):
• Try ( m = 10 ), ( k = 13 ) (( 13/10 = 1.3 ), close to 1.313):
• ( E_f = \frac{938.272}{10} = 93.8272 , \text{MeV} )
• Proton: ( 10 \cdot 93.8272 = 938.272 , \text{MeV} ) (exact)
• Delta: ( 13 \cdot 93.8272 = 1219.754 , \text{MeV} ), difference = ( 1232 - 1219.754 = 12.246 , \text{MeV} )
• N(1440): ( 15 \cdot 93.8272 = 1407.408 , \text{MeV} ), difference = ( 1440 - 1407.408 = 32.592 , \text{MeV} )
• ( 16 \cdot 93.8272 = 1501.235 , \text{MeV} ), difference = ( 1501.235 - 1440 = 61.235 , \text{MeV} ) This ( E_f = 93.8272 , \text{MeV} ) gives a closer match to the Delta (12.246 MeV off) than Option 1, while N(1440) remains similar.
• Optimize further:
• Adjust ( E_f ) for Delta: ( E_f = \frac{1232}{13} \approx 94.7692 , \text{MeV} )
• Delta: ( 13 \cdot 94.7692 = 1232 , \text{MeV} ) (exact)
• Proton: ( 10 \cdot 94.7692 = 947.692 , \text{MeV} ), difference = ( 947.692 - 938.272 = 9.42 , \text{MeV} )
• N(1440): ( 15 \cdot 94.7692 = 1421.538 , \text{MeV} ), difference = ( 1440 - 1421.538 = 18.462 , \text{MeV} ) This improves the N(1440) match but slightly offsets the proton.

Finding a single ( E_f ) that makes all three exact is challenging due to their incommensurate ratios, but ( E_f \approx 93.8272 , \text{MeV ) balances the matches well.

Decision and Spectral Plot

The query specifies “use 1/4 the proton’s energy and redo or choose an optimum fundamental.” Since “or” suggests a choice, and given the context of harmonic matching, Option 1 (( E_f = 234.568 , \text{MeV} )) adheres strictly to the instruction, placing the proton exactly at ( n = 4 ). However, Option 2 provides better overall alignment for the resonances. I’ll proceed with Option 1 as it directly follows the first directive, then note the alternative.

• Fundamental energy: ( E_f = 234.568 , \text{MeV} )
• Harmonics and amplitudes (using ( A_n = 1/n ), mimicking a bell’s spectrum):
• ( n = 1 ): 234.568 MeV, ( A = 1.000 )
• ( n = 2 ): 469.136 MeV, ( A = 0.500 )
• ( n = 3 ): 703.704 MeV, ( A = 0.333 )
• ( n = 4 ): 938.272 MeV (proton), ( A = 0.250 )
• ( n = 5 ): 1172.84 MeV, ( A = 0.200 )
• ( n = 6 ): 1407.408 MeV, ( A = 0.167 )
• ( n = 7 ): 1641.976 MeV, ( A = 0.143 )
• ( n = 8 ): 1876.544 MeV, ( A = 0.125 )
• ( n = 9 ): 2111.112 MeV, ( A = 0.111 )
• ( n = 10 ): 2345.68 MeV, ( A = 0.100 )
• Spectral plot configuration (Chart.js bar chart):

{

  "type": "bar",

  "data": {

    "labels": [234.6, 469.1, 703.7, 938.3, 1172.8, 1407.4, 1642.0, 1876.5, 2111.1, 2345.7],

    "datasets": [{

      "label": "Harmonic Energies",

      "data": [1.000, 0.500, 0.333, 0.250, 0.200, 0.167, 0.143, 0.125, 0.111, 0.100],

      "backgroundColor": "blue"

    }]

  },

  "options": {

    "scales": {

      "x": { "title": { "display": true, "text": "Energy (MeV)" } },

      "y": { "title": { "display": true, "text": "Amplitude" } }

    }

  }

}

• Resonance positions:
• Proton: Exactly at ( n = 4 ) (938.272 MeV).
• Delta (1232 MeV): Between ( n = 5 ) (1172.84 MeV, -59.16 MeV) and ( n = 6 ) (1407.408 MeV, +175.408 MeV).
• N(1440) (1440 MeV): Near ( n = 6 ) (1407.408 MeV, +32.592 MeV).

Conclusion

Using ( E_f = 234.568 , \text{MeV} ) (1/4 the proton’s energy), the proton aligns perfectly at the fourth harmonic, while the Delta and N(1440) resonances are approximate matches at higher harmonics. Alternatively, an optimum ( E_f \approx 93.8272 , \text{MeV} ) improves the Delta match (12.246 MeV off at ( n = 13 )) with the proton exact at ( n = 10 ), but since the problem emphasizes 1/4 the proton’s energy, the fundamental energy is 234.568 MeV, with the spectral plot as described.