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Imagine that a particle is moving on a ring, in the xy- plane as shown at the right. The mass of the particle is m and the radius of the ring is r. The particle is constrained on the ring, as the potential energy on the ring is assumed to be zero (V = 0). Since movement of the particle is described by two dimensions, i.e., x and y or r and p, we expect the kinetic energy term to have two terms. Use this description of the system to help you complete the questions below. 1. Identify the classic potential for a particle-on-a-ring. What coordinate system is the most reasonable choice for this problem? 2. Now that we have our potential and a coordinate system: what do we need to determine next? (Hint: the potential energy operator, see Appendix at the end of the handout for potentially useful expressions). 3. You should now have all the necessary elements to write the Schrodinger equation for a particle-on-a-ring. a. First, show/argue that for a fixed radius, r, the Hamiltonian can be expressed as: A = ↑ + V = where the moment of inertia (1) is equal to: I = mr² 21E ħ² 1 8² ħ² 2² 2m r2 ðф² 21 0Ф2 2² a6² m == b. Using the Hamiltonian, rearrange Schrödinger's equation for a particle-on-a-ring to get: 21E ħ² Y (0) = Aeim where m 4. Assuming that m² = (where m/ is a quantum number and has nothing to do with mass), show that the following is a solution to the Schrodinger equation you obtained. h²> = 21E ħ²

Imagine that a particle is moving on a ring, in the xy- plane as shown at the right. The mass of the particle is m and the radius of the ring is r. The particle is constrained on the ring, as the potential energy on the ring is assumed to be zero (V = 0). Since movement of the particle is described by two dimensions, i.e., x and y or r and p, we expect the kinetic energy term to have two terms. Use this description of the system to help you complete the questions below. 1. Identify the classic potential for a particle-on-a-ring. What coordinate system is the most reasonable choice for this problem? 2. Now that we have our potential and a coordinate system: what do we need to determine next? (Hint: the potential energy operator, see Appendix at the end of the handout for potentially useful expressions). 3. You should now have all the necessary elements to write the Schrodinger equation for a particle-on-a-ring. a. First, show/argue that for a fixed radius, r, the Hamiltonian can be expressed as: A = ↑ + V = where the moment of inertia (1) is equal to: I = mr² 21E ħ² 1 8² ħ² 2² 2m r2 ðф² 21 0Ф2 2² a6² m == b. Using the Hamiltonian, rearrange Schrödinger's equation for a particle-on-a-ring to get: 21E ħ² Y (0) = Aeim where m 4. Assuming that m² = (where m/ is a quantum number and has nothing to do with mass), show that the following is a solution to the Schrodinger equation you obtained. h²> = 21E ħ²

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Imagine that a particle is moving on a ring, in the xy- plane as shown at the right. The mass of the particle is m and the radius of the ring is r. The particle is constrained on the ring, as the potential energy on the ring is assumed to be zero (V = 0). Since movement of the particle is described by two dimensions, i.e., x and y or r and p, we expect the kinetic energy term to have two terms. Use this description of the system to help you complete the questions below. 1. Identify the classic potential for a particle-on-a-ring. What coordinate system is the most reasonable choice for this problem? 2. Now that we have our potential and a coordinate system: what do we need to determine next? (Hint: the potential energy operator, see Appendix at the end of the handout for potentially useful expressions). 3. You should now have all the necessary elements to write the Schrodinger equation for a particle-on-a-ring. a. First, show/argue that for a fixed radius, r, the Hamiltonian can be expressed as: A = ↑ + V: ħ² 1 a² 2m r² ap² ħ² 2² 21 0Ф2 where the moment of inertia (I) is equal to: I = mr² 2²4 ap² = 21E ħ² -4 Q - b. Using the Hamiltonian, rearrange Schrödinger's equation for a particle-on-a-ring to get: m where m² 21E 4. Assuming that m² = , (where my is a quantum number and has nothing to do with mass), show that the following is a solution to the Schrodinger equation you obtained. 42, (p) = A e im₁ = 21E ħ²
Transcribed Image Text:Imagine that a particle is moving on a ring, in the xy- plane as shown at the right. The mass of the particle is m and the radius of the ring is r. The particle is constrained on the ring, as the potential energy on the ring is assumed to be zero (V = 0). Since movement of the particle is described by two dimensions, i.e., x and y or r and p, we expect the kinetic energy term to have two terms. Use this description of the system to help you complete the questions below. 1. Identify the classic potential for a particle-on-a-ring. What coordinate system is the most reasonable choice for this problem? 2. Now that we have our potential and a coordinate system: what do we need to determine next? (Hint: the potential energy operator, see Appendix at the end of the handout for potentially useful expressions). 3. You should now have all the necessary elements to write the Schrodinger equation for a particle-on-a-ring. a. First, show/argue that for a fixed radius, r, the Hamiltonian can be expressed as: A = ↑ + V: ħ² 1 a² 2m r² ap² ħ² 2² 21 0Ф2 where the moment of inertia (I) is equal to: I = mr² 2²4 ap² = 21E ħ² -4 Q - b. Using the Hamiltonian, rearrange Schrödinger's equation for a particle-on-a-ring to get: m where m² 21E 4. Assuming that m² = , (where my is a quantum number and has nothing to do with mass), show that the following is a solution to the Schrodinger equation you obtained. 42, (p) = A e im₁ = 21E ħ²
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