The problem you're describing deals with the quantum harmonic oscillator, which is one of the most fundamental systems in quantum mechanics. The potential given is $V(x) = \frac{1}{2} m \omega^2 x^2$, representing a harmonic oscillator with mass $m$ and angular frequency $\omega$. We aim to determine the energy levels of this system using the time-independent Schrödinger equation.

Step 1: The Schrödinger Equation

The time-independent Schrödinger equation for a system is given by: $\hat{H} \psi(x) = E \psi(x)$ where $\hat{H}$ is the Hamiltonian operator. For the quantum harmonic oscillator, the Hamiltonian is the sum of the kinetic and potential energy operators: $\hat{H} = -\frac{\hbar^2}{2m} \frac{d^2}{dx^2} + \frac{1}{2} m \omega^2 x^2$ Here, $\hbar$ is the reduced Planck constant, and $\omega$ is the angular frequency of the oscillator.

Step 2: Energy Levels

By solving the Schrödinger equation for this potential, we find that the energy eigenvalues $E_n$ are quantized and given by the formula: $E_n = \left( n + \frac{1}{2} \right) \hbar \omega \quad \text{where} \quad n = 0, 1, 2, 3, \dots$ This is the general expression for the energy of the quantum harmonic oscillator. The term $\frac{1}{2} \hbar \omega$ represents the zero-point energy, which is the energy the system has even in its ground state ($n = 0$).

Step 3: First Three Energy Levels

To determine the first three energy levels explicitly, substitute $n = 0$, $n = 1$, and $n = 2$ into the general formula:

1. Ground state (n = 0): $E_0 = \left( 0 + \frac{1}{2} \right) \hbar \omega = \frac{1}{2} \hbar \omega$

2. First excited state (n = 1): $E_1 = \left( 1 + \frac{1}{2} \right) \hbar \omega = \frac{3}{2} \hbar \omega$

3. Second excited state (n = 2): $E_2 = \left( 2 + \frac{1}{2} \right) \hbar \omega = \frac{5}{2} \hbar \omega$

Step 4: General Expression for Energy Levels

The general expression for the energy levels of the quantum harmonic oscillator is: $E_n = \left( n + \frac{1}{2} \right) \hbar \omega$ This formula applies to all $n = 0, 1, 2, 3, \dots$, where $n$ is the quantum number representing the energy level.

Summary:

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