2. For the following 4 cases, set up the correct integral to find the expectation values, for example (P.) = sin(-ih) sin()) dx. fª (²3) ² (a) for the second excited state of a particle in a 1D box of width a (b) for the first excited state of a harmonic oscillator 1/2 (2²) ¹/² (sin 8) ei (c) for the spherical harmonic Y₁¹ TE Ly in spherical coordinates as Ә Ly = -ih (cosp cot 0 sin ) ae (d) for the radial portion of a ground state (1s) electron in a hydrogen atom
2. For the following 4 cases, set up the correct integral to find the expectation values, for example (P.) = sin(-ih) sin()) dx. fª (²3) ² (a) for the second excited state of a particle in a 1D box of width a (b) for the first excited state of a harmonic oscillator 1/2 (2²) ¹/² (sin 8) ei (c) for the spherical harmonic Y₁¹ TE Ly in spherical coordinates as Ә Ly = -ih (cosp cot 0 sin ) ae (d) for the radial portion of a ground state (1s) electron in a hydrogen atom
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![### Expectation Values in Quantum Mechanics
For the following four cases, we will set up the correct integral to find the expectation values. Here’s an example of how to find \(\langle p_x \rangle\):
\[
\langle p_x \rangle = \int_0^a \left(\frac{2}{a}\right)^{\frac{1}{2}} \sin \left(\frac{\pi x}{a}\right) \left(-i \hbar \frac{\partial}{\partial x} \left[ \left(\frac{2}{a}\right)^{\frac{1}{2}} \sin \left(\frac{\pi x}{a}\right) \right] \right) dx
\]
#### Cases:
- **(a)** \(\langle x^2 \rangle\) for the second excited state of a particle in a 1D box of width \(a\).
- **(b)** \(\langle x \rangle\) for the first excited state of a harmonic oscillator.
- **(c)** \(\langle L_y \rangle\) for the spherical harmonic \(Y_1^1 = \left(\frac{3}{8\pi}\right)^{1/2} (\sin \theta) e^{i\phi}\).
In spherical coordinates, the operator \(\hat{L_y}\) is defined as:
\[
\hat{L_y} = -i\hbar \left( \cos \phi \frac{\partial}{\partial \theta} - \cot \theta \sin \phi \frac{\partial}{\partial \phi} \right)
\]
- **(d)** \(\langle r \rangle\) for the radial portion of a ground state (1s) electron in a hydrogen atom.
Note: Certain lines have been redacted for clarity.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff6ab806b-464c-4756-9d8f-72da0489465e%2F8b7f90a3-842c-46ed-b668-284cbdd4d023%2F2k9mvld_processed.png&w=3840&q=75)
Transcribed Image Text:### Expectation Values in Quantum Mechanics
For the following four cases, we will set up the correct integral to find the expectation values. Here’s an example of how to find \(\langle p_x \rangle\):
\[
\langle p_x \rangle = \int_0^a \left(\frac{2}{a}\right)^{\frac{1}{2}} \sin \left(\frac{\pi x}{a}\right) \left(-i \hbar \frac{\partial}{\partial x} \left[ \left(\frac{2}{a}\right)^{\frac{1}{2}} \sin \left(\frac{\pi x}{a}\right) \right] \right) dx
\]
#### Cases:
- **(a)** \(\langle x^2 \rangle\) for the second excited state of a particle in a 1D box of width \(a\).
- **(b)** \(\langle x \rangle\) for the first excited state of a harmonic oscillator.
- **(c)** \(\langle L_y \rangle\) for the spherical harmonic \(Y_1^1 = \left(\frac{3}{8\pi}\right)^{1/2} (\sin \theta) e^{i\phi}\).
In spherical coordinates, the operator \(\hat{L_y}\) is defined as:
\[
\hat{L_y} = -i\hbar \left( \cos \phi \frac{\partial}{\partial \theta} - \cot \theta \sin \phi \frac{\partial}{\partial \phi} \right)
\]
- **(d)** \(\langle r \rangle\) for the radial portion of a ground state (1s) electron in a hydrogen atom.
Note: Certain lines have been redacted for clarity.
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