Time independent Schrödinger Eqn in a harmonic potential

In summary, the conversation discusses the process of solving the Schrödinger equation for the harmonic oscillator using the series method. It involves using two dimensionless variable substitutions and the chain rule to simplify the equation. The person is seeking clarification on how to obtain the final equation.
  • #1
Old_sm0key
17
0

Homework Statement


I am currently reading a textbook on solving the Schrödinger equation for the harmonic oscillator using the series method;
$$-\frac{\hbar^{2}}{2m}\frac{\mathrm{d}^2 \psi }{\mathrm{d} x^2}+\frac{1}{2}m\omega ^{2}x^2\psi =E\psi $$

It starts by using these two dimensionless variable substitutions (which I gather is standard practise): $$\xi \equiv \sqrt{\frac{m\omega }{\hbar}}x$$and$$K\equiv \frac{2E}{\hbar\omega }$$
to produce the simplified equation: $$\frac{\mathrm{d} ^2\psi }{\mathrm{d} \xi ^2}=\left ( \xi ^2-K \right )\psi $$

I cannot match the final equation using these substitutions alone. Surely there must be some adjustment for the change of variables in the derivative? Please can someone explain how to get the final (simplified) equation? (I am a newcomer to quantum mechanics!)

Thanks in advance.
 
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  • #2
You need to use the chain rule: ##\frac{d}{dx} = \frac{d\xi}{dx} \frac{d}{d\xi}##.
 

1. What is the Time Independent Schrödinger Equation?

The Time Independent Schrödinger Equation is a fundamental equation in quantum mechanics that describes the behavior of a quantum system in a stationary state. It is used to calculate the wave function, which describes the probability of finding a particle at a certain location in space.

2. What is a harmonic potential?

A harmonic potential is a type of potential energy function that is used to describe the force acting on a particle in a simple harmonic oscillator. It is a quadratic function that increases proportionally to the square of the distance from the equilibrium position.

3. How is the Time Independent Schrödinger Equation used in a harmonic potential?

In a harmonic potential, the Time Independent Schrödinger Equation is used to find the allowed energy levels and corresponding wave functions for a particle in a simple harmonic oscillator. The equation is solved for the specific potential energy function, and the solutions give the energy eigenvalues and eigenfunctions for the system.

4. What is the significance of the solutions to the Time Independent Schrödinger Equation in a harmonic potential?

The solutions to the Time Independent Schrödinger Equation in a harmonic potential represent the quantized energy levels and corresponding wave functions for a particle in a simple harmonic oscillator. These solutions are important in understanding the behavior of systems in quantum mechanics and have applications in various fields such as quantum chemistry and solid-state physics.

5. Are there any limitations to using the Time Independent Schrödinger Equation in a harmonic potential?

While the Time Independent Schrödinger Equation is a powerful tool in quantum mechanics, it is limited to systems that are in a stationary state. This means that it cannot be used to describe systems that are changing over time, or those that are in non-equilibrium states. Additionally, the equation assumes a perfect harmonic potential, which may not accurately describe all physical systems.

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