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Problem 3. Consider the two example systems from quantum mechanics. First, for a particle in a box of length 1 we have the equation h² d²v = EV, 2m dr² with boundary conditions (0) = 0 and V(1) = 0. Second, the Quantum Harmonic Oscillator (QHO) h² d² 2m dz² +kr² + 1/ k₂²) v = EV (a) Write down the states for both systems. What are their similarities and differences? (b) Write down the energy eigenvalues for both systems. What are their similarities and differences? (c) Plot the first three states of the QHO along with the potential for the system.

Problem 3. Consider the two example systems from quantum mechanics. First, for a particle in a box of length 1 we have the equation h² d²v = EV, 2m dr² with boundary conditions (0) = 0 and V(1) = 0. Second, the Quantum Harmonic Oscillator (QHO) h² d² 2m dz² +kr² + 1/ k₂²) v = EV (a) Write down the states for both systems. What are their similarities and differences? (b) Write down the energy eigenvalues for both systems. What are their similarities and differences? (c) Plot the first three states of the QHO along with the potential for the system.

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**Problem 3.** Consider the two example systems from quantum mechanics. First, for a particle in a box of length 1 we have the equation \[ -\frac{\hbar^2}{2m} \frac{d^2 \Psi}{dx^2} = E \Psi, \] with boundary conditions \(\Psi(0) = 0\) and \(\Psi(1) = 0\). Second, the Quantum Harmonic Oscillator (QHO) \[ \left( -\frac{\hbar^2}{2m} \frac{d^2}{dx^2} + \frac{1}{2} kx^2 \right) \Psi = E \Psi \] (a) Write down the states for both systems. What are their similarities and differences? (b) Write down the energy eigenvalues for both systems. What are their similarities and differences? (c) Plot the first three states of the QHO along with the potential for the system. (d) Explain why you can observe a particle outside of the “classically allowed region”. *Hint: you can use any state and compute an integral to determine a probability of a particle being in a given region.*
Transcribed Image Text:**Problem 3.** Consider the two example systems from quantum mechanics. First, for a particle in a box of length 1 we have the equation \[ -\frac{\hbar^2}{2m} \frac{d^2 \Psi}{dx^2} = E \Psi, \] with boundary conditions \(\Psi(0) = 0\) and \(\Psi(1) = 0\). Second, the Quantum Harmonic Oscillator (QHO) \[ \left( -\frac{\hbar^2}{2m} \frac{d^2}{dx^2} + \frac{1}{2} kx^2 \right) \Psi = E \Psi \] (a) Write down the states for both systems. What are their similarities and differences? (b) Write down the energy eigenvalues for both systems. What are their similarities and differences? (c) Plot the first three states of the QHO along with the potential for the system. (d) Explain why you can observe a particle outside of the “classically allowed region”. *Hint: you can use any state and compute an integral to determine a probability of a particle being in a given region.*
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