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 dx² with boundary conditions (0) = 0 and V(1) = 0. Second, the Quantum Harmonic Oscillator (QHO) = h² d² +kr²V = EV 2m dg²+ka² 1/ k2²) v (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.
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 dx² with boundary conditions (0) = 0 and V(1) = 0. Second, the Quantum Harmonic Oscillator (QHO) = h² d² +kr²V = EV 2m dg²+ka² 1/ k2²) v (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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Question
![**Problem 3.** Consider the two example systems from quantum mechanics. First, for a particle in a box of length 1 we have the equation
\[
-\dfrac{\hbar^2}{2m} \dfrac{d^2\Psi}{dx^2} = E\Psi,
\]
with boundary conditions \(\Psi(0) = 0\) and \(\Psi(1) = 0\).
Second, the Quantum Harmonic Oscillator (QHO)
\[
\left( -\dfrac{\hbar^2}{2m} \dfrac{d^2}{dx^2} + \dfrac{1}{2} k x^2 \right) \Psi = E \Psi
\]
(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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7bf14c43-8cf8-40bd-94f2-90056f6925f0%2F96355a81-faad-42b3-b96f-2b909e0a328f%2Fkgwo2dm_processed.png&w=3840&q=75)
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
\[
-\dfrac{\hbar^2}{2m} \dfrac{d^2\Psi}{dx^2} = E\Psi,
\]
with boundary conditions \(\Psi(0) = 0\) and \(\Psi(1) = 0\).
Second, the Quantum Harmonic Oscillator (QHO)
\[
\left( -\dfrac{\hbar^2}{2m} \dfrac{d^2}{dx^2} + \dfrac{1}{2} k x^2 \right) \Psi = E \Psi
\]
(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.
Expert Solution
Step 1
Given the length of 1 Dimension box is 1.
And given a 1 dimension hormonic oscillator. Let the mass of particle is m which is moving in these potential.
Important thing this that the difference between energy level is same for 1 dimension hormonic oscillator for all n that is they are equidistant but in box it is different for all value of n.
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