Quantum harmonic oscillator 1d

In summary, the conversation discusses how to derive a version of the Schrodinger Equation for QHO using the TISE and a given potential. To do so, equation 4 is obtained using the chain rule for derivatives and equations 1 and 5 are used to replace variables in equation 3. Further assistance is requested on how to acquire equation 4.
  • #1
1
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Ok, so I am trying understand how to derive the following version of the Schrodinger Equation for QHO:

[tex]\frac{d^2u}{dz^2} + (2\epsilon-z^2)u=0[/tex]

where

[tex]\ 1. z=(\frac{m\omega}{hbar})^{1/2}x [/tex] and

[tex]\ 2. \epsilon= \frac{E}{hbar\omega}[/tex]

I've started with the TISE, used a potential of V(x)=1/2mw^2x^2, and with a little rearranging have the following:

[tex]\ 3. \frac{d^2u}{dx^2} + \frac{2mE}{hbar^2}u - \frac{m^2\omega^2x^2}{hbar^2}u = 0 [/tex]

Using equation 1 we have:

[tex]\ 4. \frac{d^2u}{dx^2} = \frac{m\omega}{hbar}\frac{d^2u}{dz^2}[/tex]

and

[tex]\ 5.x^2=\frac{hbar}{m\omega}z^2[/tex]

Plugging 4 and 5 into equation 3 and rearranging gives the required answer.

I can follow all of the steps, but get stuck on how to acquire the equation 4. Any help would be much appreciated.
 
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  • #2
rooney123 said:
Ok, so I am trying understand how to derive the following version of the Schrodinger Equation for QHO:

[tex]\frac{d^2u}{dz^2} + (2\epsilon-z^2)u=0[/tex]

where

[tex]\ 1. z=(\frac{m\omega}{hbar})^{1/2}x [/tex] and

[tex]\ 2. \epsilon= \frac{E}{hbar\omega}[/tex]

I've started with the TISE, used a potential of V(x)=1/2mw^2x^2, and with a little rearranging have the following:

[tex]\ 3. \frac{d^2u}{dx^2} + \frac{2mE}{hbar^2}u - \frac{m^2\omega^2x^2}{hbar^2}u = 0 [/tex]

Using equation 1 we have:

[tex]\ 4. \frac{d^2u}{dx^2} = \frac{m\omega}{hbar}\frac{d^2u}{dz^2}[/tex]

and

[tex]\ 5.x^2=\frac{hbar}{m\omega}z^2[/tex]

Plugging 4 and 5 into equation 3 and rearranging gives the required answer.

I can follow all of the steps, but get stuck on how to acquire the equation 4. Any help would be much appreciated.

Chain rule for derivatives: for f(v) and v(y), df/dy=(df/dv)(dv/dy) (the variables have been renamed to protect the innocent :wink:).

Is that enough of a hint, or do you need more help?
 

1. What is a quantum harmonic oscillator 1d?

A quantum harmonic oscillator is a system in quantum mechanics that describes the motion of a particle that is confined to a potential well. In one dimension, the particle can only move along a straight line.

2. How is the energy of a quantum harmonic oscillator 1d calculated?

The energy of a quantum harmonic oscillator 1d is calculated using the Schrödinger equation, which is a mathematical equation that describes the behavior of quantum systems. The energy is quantized, meaning it can only take on certain discrete values.

3. What is the significance of the ground state in a quantum harmonic oscillator 1d?

The ground state is the lowest energy state of a quantum harmonic oscillator. It has important implications in various fields of physics, including quantum mechanics and statistical mechanics. In many systems, the ground state is the state in which the system is most likely to be found.

4. How does the potential energy affect the behavior of a quantum harmonic oscillator 1d?

The potential energy plays a crucial role in determining the behavior of a quantum harmonic oscillator 1d. It determines the allowed energy levels and the probability of the particle being in a certain state. A steeper potential well results in a larger energy gap between levels, while a shallower potential well allows for more energy states.

5. What are some real-world applications of the quantum harmonic oscillator 1d?

The quantum harmonic oscillator 1d has many applications in fields such as quantum computing, quantum chemistry, and quantum optics. It is also used to describe the behavior of atoms, molecules, and other microscopic systems. In addition, it has been used to model the behavior of financial markets and biological systems.

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