- Solve 5 And 6 1 (95.53 KiB) Viewed 48 times
Solve (5) and (6)
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answerhappygod
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Solve (5) and (6)
Solve (5) and (6)
5. A Step Up on the Infinite Line [10 points] A particle of mass m is moving in one dimension, subject to the potential V(x): V(x) = Vo, 0, for x > 0, for x ≤0. Find the stationary states that exist for energies 0<E< Vo. 6. A Wall and Half of a Finite Well [10 points] A particle of mass m is moving in one dimension, subject to the potential V(x): ∞, for r < 0, V(x)=-Vo, for 0<x<a, (Vo > 0) for x > a. 0, Find the stationary states that correspond to bound states (E< 0, in this case). Is there always a bound state? Find the minimum value of zo 2ma²Vo ħ² for which there are three bound states. Explain the precise relation of this problem to the problem of the finite square well of width 2a. 7. Mimicking hydrogen with a one-dimensional square well. [5 points] The hydrogen atom the Bohr radius ao and ground state energy Eo are given by ħ² p² ao = ~0.529 x 10-¹0m, Eo - = -13.6 eV. me² 200 The ground state is a bound state and the potential goes to zero at infinity. We want to design a one-dimensional finite square well -Vo, for a <ao, Vo > 0, 0, for x> ao V(x) = that simulates the hydrogen atom. Calculate the value of Vo in eV so that the ground state of the box is at the correct depth. 8. No states with E<V(x) [5 points] Consider a real stationary state (r) with energy E: ħ² -"(x) + [V(x) - Ev(x) = 0. 2m (a) Prove that E must exceed the minimum value of V(r) by noting that E = (H). (b) Explain the claim by trying (and failing) to sketch a wavefunction consistent with being on the classically inaccessible region for all values of x.
5. A Step Up on the Infinite Line [10 points] A particle of mass m is moving in one dimension, subject to the potential V(x): V(x) = Vo, 0, for x > 0, for x ≤0. Find the stationary states that exist for energies 0<E< Vo. 6. A Wall and Half of a Finite Well [10 points] A particle of mass m is moving in one dimension, subject to the potential V(x): ∞, for r < 0, V(x)=-Vo, for 0<x<a, (Vo > 0) for x > a. 0, Find the stationary states that correspond to bound states (E< 0, in this case). Is there always a bound state? Find the minimum value of zo 2ma²Vo ħ² for which there are three bound states. Explain the precise relation of this problem to the problem of the finite square well of width 2a. 7. Mimicking hydrogen with a one-dimensional square well. [5 points] The hydrogen atom the Bohr radius ao and ground state energy Eo are given by ħ² p² ao = ~0.529 x 10-¹0m, Eo - = -13.6 eV. me² 200 The ground state is a bound state and the potential goes to zero at infinity. We want to design a one-dimensional finite square well -Vo, for a <ao, Vo > 0, 0, for x> ao V(x) = that simulates the hydrogen atom. Calculate the value of Vo in eV so that the ground state of the box is at the correct depth. 8. No states with E<V(x) [5 points] Consider a real stationary state (r) with energy E: ħ² -"(x) + [V(x) - Ev(x) = 0. 2m (a) Prove that E must exceed the minimum value of V(r) by noting that E = (H). (b) Explain the claim by trying (and failing) to sketch a wavefunction consistent with being on the classically inaccessible region for all values of x.