Question

Harmonic Oscillator: The hannonic oscillator can be solved exactly for the quantized energies, and here we compare those results with a numerical approach. Along with the values of \(m\) and \(\hbar\) discussed above, we choose our units so that the spring constant \(x\) is also 1 . The potential energy function \(U(x)\) is then simply \(\frac{1}{2} x^{2}\) For \(\Delta x\), use \(0.001\). (a) Following the above guidelines on choosing \(1 \beta(0)\) and \(\psi(\Delta x)\), test both odd and even functions at different trial values of \(E\) by finding \(\psi\) at all positive multiples of \(\Delta x\) out \(10 x=4\) and plotting the results. Note that because of the functions' symmetries there is no need to plot negative values of \(x\). Find four allowed energies. (b) What tells you that an energy is correct? (c) Compare your results with the exact values given in equation \((5-26)\).

Short answer

The energy levels can be quantitatively decided by observing the conditions on wave functions. Four correct energy levels that satisfy Schrödinger's equation for the harmonic oscillator are determined through this problem. These are then compared to the exact solutions from equation (5-26).

Step-by-step solution

Preparation

First, understand what the harmonic oscillator is and acquainted with all the variables used in this problem. Also, understand the variables that have been set to unity for simplicity.

Apply the Finite Difference Method

Following the guidelines, create an algorithm to solve the Schroedinger equation using the finite difference method. Choose different trial energy values \(E\) and compute \(\psi(x)\) for all positive multiples of \(\Delta x\) up to \(x = 4\). Use different initial values for \(\beta(0)\) and \(\psi(\Delta x)\) as discussed.

Plot the results

Plot the results from the different trial energy functions. Four plots will be created - for four different trial energies.

Determine correct Energy

A possible trial energy is correct if the function \(\psi(x)\) approaches zero as \(x\) approaches infinity. This has to be checked for each plot.

Compare with exact solutions

Compare the approximate energies obtained from the graphical solution with the exact energy levels given by equation (5-26).

Key concepts

Quantum Mechanics

Quantum mechanics is an essential part of modern physics that helps us understand the nature of matter and energy on very small scales, such as atoms and subatomic particles. Unlike classical mechanics, quantum mechanics suggests that particles can exist in multiple states at the same time until a measurement is made. This is known as superposition.
One of the core ideas in quantum mechanics is quantization, which refers to the discrete (quantized) nature of many physical properties like energy, charge, and angular momentum. For example, electrons in an atom can only exist in certain allowed energy levels, and they cannot exist in between them.
In the context of the harmonic oscillator, a popular model in quantum mechanics, the energies are also quantized, which means the oscillator can only occupy specific energy levels. Understanding these ideas helps explain a great deal of phenomena, such as why atoms emit or absorb only specific frequencies of light.

Schroedinger Equation

The Schroedinger Equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. It’s like Newton’s second law in classical mechanics, but for quantum systems. The equation is expressed as:\[i \; i \hbar \frac{\partial}{\partial t} \psi(x,t) = \hat{H} \psi(x,t) \; \], where \( \hat{H} \) is the Hamiltonian operator, \( \hbar \) is the reduced Planck's constant, and \( \psi(x,t) \) is the wavefunction of the system.
For a harmonic oscillator, which is a system where a particle experiences a force proportional to its displacement from a fixed point, this equation can be simplified for stationary states as:\[\hat{H} \psi(x) = E \psi(x) \] where \( E \) represents the energy of the state.
The solutions to this equation provide us with the possible wavefunctions \( \psi(x) \) that describe the probability of finding a particle at a specific position \( x \). This is crucial for understanding how particles behave under different potential energies, like the quadratic potential in a harmonic oscillator.

Finite Difference Method

The Finite Difference Method is a numerical approach used to solve differential equations, like the Schroedinger equation, that can be quite complex to solve analytically. This method approximates derivatives by using differences between function values at certain points.
To apply this method, the continuous space is divided into discrete points spaced by a small distance \( \Delta x \). A function such as \( \psi(x) \), the wavefunction in quantum mechanics, is calculated at these discrete points.
By choosing trial energy values, the Finite Difference Method allows us to compute approximate values for \( \psi(x) \) using initial guesses. The result is often a numerical plot that helps us visualize where the wavefunction reaches zero, which indicates a correct energy level as noted in our problem's solution.
This numerical method is especially useful when an exact analytical solution is difficult to find or doesn’t exist.

Energy Levels

Energy levels in quantum systems like the harmonic oscillator are discrete values that a particle can occupy as a result of quantization. In classical physics, a harmonic oscillator could have any energy. But, in the quantum world, it can only have specific energy levels given by:\[ E_n = \hbar \omega\left(n + \frac{1}{2}\right) \; \]where \( n \) is a non-negative integer (0, 1, 2,...) and \( \omega \) is the angular frequency of the oscillator.
Finding these quantized energy levels is essential for understanding how quantum systems like molecules and atoms work. This is because transitions between these levels correspond to absorption or emission of photons with specific energies, which is a fundamental aspect of spectroscopy.
In numerical simulations, these energy levels can be explored using methods like the Finite Difference Method to find how closely numerically derived energies match the exact values from quantum theory.