: The ground state eigenfunction for a linear harmonic oscillator, in terms of a=\sqrt{\frac{mk}{n^{2}}} where k = force constant and m = mass of the linear oscillator, is

(α) ψ(x) 1/4 eax2/2

(9) \psi(x)=(\frac{a}{\pi})^{1/4}e^{-ax^{2}/2}

(c) (x) (元) = 2

(d) \psi(x)=(\frac{a}{\sqrt{\pi}})^{1/6}e^{+a^{2}x^{2}/2}

The correct expression for the ground state eigenfunction \( \psi(x) \) of a linear harmonic oscillator is:

\[
\psi(x) = \left(\frac{a}{\pi}\right)^{1/4} e^{-a x^2 / 2}
\]

This is option (b).

### Explanation:
1. The **ground state eigenfunction** for a one-dimensional quantum harmonic oscillator is generally written as:

   \[
   \psi_0(x) = \left( \frac{m \omega}{\pi \hbar} \right)^{1/4} e^{- \frac{m \omega}{2 \hbar} x^2}
   \]
   
   Here, \( m \) is the mass, \( \omega \) is the angular frequency, and \( \hbar \) is the reduced Planck constant.

2. In the problem, the parameter \( a \) is defined as \( a = \sqrt{\frac{mk}{\hbar^2}} \), where \( k \) is the force constant. This gives the simplified form of the ground state eigenfunction:

   \[
   \psi(x) = \left( \frac{a}{\pi} \right)^{1/4} e^{-a x^2 / 2}
   \]

This matches option (b).

Would you like to go over details or ask any further questions?

### Related Questions:
1. How is the ground state energy of the harmonic oscillator calculated?
2. What is the general form of the eigenfunctions for excited states of a harmonic oscillator?
3. How does the force constant \( k \) affect the frequency and energy levels of the oscillator?
4. What is the role of the parameter \( a \) in the wavefunction's shape and normalization?
5. How is the expectation value of position calculated for the ground state?

### Tip:
In quantum mechanics, the ground state wavefunction is always real and normalized, and it describes the lowest energy state of the system with no nodes.

The ground state eigenfunction for a linear harmonic oscillator, in terms of a=\sqrt{\frac{mk}{n^{2}}} where k = force constant and m = mass of the linear oscillator, is: (a) ψ(x) 1/4 e^{ax^2/2}, (b) ψ(x)=(\frac{a}{\pi})^{1/4}e^{-ax^{2}/2}, (c) ψ(x) 2, (d) ψ(x)=(\frac{a}{\sqrt{\pi}})^{1/6}e^{+a^{2}x^{2}/2}.

Explore the ground state eigenfunction of a linear harmonic oscillator in quantum mechanics, defined as ψ(x) = (\frac{a}{\pi})^{1/4} e^{-a x^2 / 2}. Learn about wavefunctions, eigenfunctions, and the quantum harmonic oscillator theorem, suitable for undergraduate physics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation