Consider the wave function for the ground state harmonic oscillator: x) = ("nh 1/4 A. What is the quantum number for this ground state? v = B. Enter the integrand you'd need to evaluate (x) for the ground state harmonic oscillator wave function: (x) - dx C. Evaluate the integral in part B. What do you obtain for the average displacement?|
Consider the wave function for the ground state harmonic oscillator: x) = ("nh 1/4 A. What is the quantum number for this ground state? v = B. Enter the integrand you'd need to evaluate (x) for the ground state harmonic oscillator wave function: (x) - dx C. Evaluate the integral in part B. What do you obtain for the average displacement?|
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![**Problem Set on Quantum Mechanics**
**1. Harmonic Oscillator**
Consider the wave function for the ground state harmonic oscillator:
\[
\psi(x) = \left( \frac{m \omega}{\pi \hbar} \right)^{1/4} e^{-m \omega x^2 / (2 \hbar)}
\]
A. What is the quantum number for this ground state? \[ \nu = \_\_\_\_\_\_\_ \]
B. Enter the integrand you’d need to evaluate \( \langle x \rangle \) for the ground state harmonic oscillator wave function:
\[
\langle x \rangle = \int_{-\infty}^{\infty} \_\_\_\_\_\_\_\_\_ \, dx
\]
C. Evaluate the integral in part B. What do you obtain for the average displacement? \[ \_\_\_\_\_\_\_ \]
**2. Particle in a Box**
Consider a particle confined to a 1-dimensional box of length \( L = 12.2 \, \text{nm} \).
A. What is the probability of locating the particle between \( x = 1.3 \, \text{nm} \) and \( x = 11.1 \, \text{nm} \) in the second excited state (Hint: n=3)? \[ \_\_\_\_\_\_\_ \]
B. Evaluate \( \hat{T} \psi \) where \( \psi \) is the normalized particle in a box wave function. \[ \_\_\_\_\_\_\_ \]
C. Evaluate \( \hat{V} \psi \) where \( \psi \) is the normalized particle in a box wave function. \[ \_\_\_\_\_\_\_ \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa0bb33da-4292-4a51-8799-113a66f1981e%2F0e378587-2a28-4de0-aaeb-c4e0825803a5%2Fby25y1e_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Set on Quantum Mechanics**
**1. Harmonic Oscillator**
Consider the wave function for the ground state harmonic oscillator:
\[
\psi(x) = \left( \frac{m \omega}{\pi \hbar} \right)^{1/4} e^{-m \omega x^2 / (2 \hbar)}
\]
A. What is the quantum number for this ground state? \[ \nu = \_\_\_\_\_\_\_ \]
B. Enter the integrand you’d need to evaluate \( \langle x \rangle \) for the ground state harmonic oscillator wave function:
\[
\langle x \rangle = \int_{-\infty}^{\infty} \_\_\_\_\_\_\_\_\_ \, dx
\]
C. Evaluate the integral in part B. What do you obtain for the average displacement? \[ \_\_\_\_\_\_\_ \]
**2. Particle in a Box**
Consider a particle confined to a 1-dimensional box of length \( L = 12.2 \, \text{nm} \).
A. What is the probability of locating the particle between \( x = 1.3 \, \text{nm} \) and \( x = 11.1 \, \text{nm} \) in the second excited state (Hint: n=3)? \[ \_\_\_\_\_\_\_ \]
B. Evaluate \( \hat{T} \psi \) where \( \psi \) is the normalized particle in a box wave function. \[ \_\_\_\_\_\_\_ \]
C. Evaluate \( \hat{V} \psi \) where \( \psi \) is the normalized particle in a box wave function. \[ \_\_\_\_\_\_\_ \]
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