Harmonic Oscillator equivalence

In summary, the conversation discusses the equation shown in section 2.2 of "Introduction to Quantum Mechanics, 2nd edition" (Griffiths) which is derived from the time-independent Schrödinger equation and involves the variable k. It is referred to as the classical simple harmonic oscillator equation, but there is no connection to the harmonic oscillator equation for masses and springs with constant k. The similarity lies in the mathematical form of the equations, but they describe different physical phenomena. The mention of the classical SHO in the book is simply a way of indicating a known equation for solving the problem.
  • #1
Adolfo Scheidt
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1
Hello, I'm studying the section 2.2 of "Introduction to Quantum Mechanics, 2nd edition" (Griffiths), and he shows this equation $$\frac{\partial^2\psi}{\partial x^2} = -k^2\psi , $$ where psi is a function only of x (this equation was derivated from the time-independent Schrödinger equation) and k is defined as $$\frac{\sqrt{2mE}}{\hbar}.$$

Then, he refers to the equation above as being the classical simple harmonic oscillator equation. That's where my question comes: I can't exactly see how he made this connection with the following harmonic oscillator equation (for masses and springs with constant k, for example) $$ m\frac{d^2 x}{dt^2} = -kx$$

Any help will be very appreciated. Thanks :)
 
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  • #2
Mathematically it is the same equation, just changing the names of the variables.
 
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  • #3
Orodruin said:
Mathematically it is the same equation, just changing the names of the variables.
Which is a misleading coincidence.
 
  • #4
Adolfo Scheidt said:
Hello, I'm studying the section 2.2 of "Introduction to Quantum Mechanics, 2nd edition" (Griffiths), and he shows this equation $$\frac{\partial^2\psi}{\partial x^2} = -k^2\psi , $$ where psi is a function only of x (this equation was derivated from the time-independent Schrödinger equation) and k is defined as $$\frac{\sqrt{2mE}}{\hbar}.$$

Then, he refers to the equation above as being the classical simple harmonic oscillator equation. That's where my question comes: I can't exactly see how he made this connection with the following harmonic oscillator equation (for masses and springs with constant k, for example) $$ m\frac{d^2 x}{dt^2} = -kx$$

Any help will be very appreciated. Thanks :)
The ##k## in the first equation (Sec. 2.2) has nothing to do with the ##k## in the last equation (Sec. 2.3). Your first and second equation have nothing to do with harmonic oscillator, and Griffiths does not say that they do.
 
  • #5
Demystifier said:
Which is a misleading coincidence.
I do not see why you want to claim that my post is misleading. It is the same differential equation. That it describes completely different things that a priori are physically different is a different matter. (Also, one comes with boundary conditions at two points and the other with initial conditions in a single point, which makes them mathematically different. But mathematically, the differential equations themselves are the same.)
 
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  • #6
Demystifier said:
The ##k## in the first equation (Sec. 2.2) has nothing to do with the ##k## in the last equation (Sec. 2.3). Your first and second equation have nothing to do with harmonic oscillator, and Griffiths does not say that they do.

It's interesting how easy it is to give a misleading impression. Why didn't Griffiths just say "and we have a well-known second-order ODE, whose solution is ..."? Why did he have to mention the classical SHO at all? I'll bet he never even thought about it. It was just a way to say "here's an equation we already know how to solve."
 
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  • #7
PeroK said:
Why did he have to mention the classical SHO at all? I'll bet he never even thought about it. It was just a way to say "here's an equation we already know how to solve."

Boundary conditions and normalization factors? For finding the set of possible solutions certainly. Much of the physics is often elsewhere.
 
  • #8
Demystifier said:
Which is a misleading coincidence.

It's NOT a coincidence, once you realize you don't need Schroedinger wavefunctions, but can do very well with Heisenberg matrices (or, if you prefer, time-dependent operators in the Heisenberg picture).
 
  • #9
Orodruin said:
Mathematically it is the same equation, just changing the names of the variables.
I was thinking about that; actually, it's the answer that makes more sense to me. I just thought that was kind of "weird" the similarity of the equations on a physical context, but if the focus is the "shape" of the equation (Mathematically), makes all sense. Thank you :)
 

Related to Harmonic Oscillator equivalence

1. What is the concept of Harmonic Oscillator equivalence?

The concept of Harmonic Oscillator equivalence refers to the principle in physics that states that any system that exhibits simple harmonic motion can be described by an equation that is mathematically equivalent to that of a simple harmonic oscillator. This means that the motion of different systems, such as a mass on a spring or a pendulum, can be described using the same mathematical equations.

2. How is Harmonic Oscillator equivalence used in science?

Harmonic Oscillator equivalence is used in various fields of science, including physics, engineering, and chemistry. It allows scientists to describe and analyze the motion of different systems using a unified mathematical framework, making it easier to compare and understand the behavior of different systems.

3. What is the equation for a simple harmonic oscillator?

The equation for a simple harmonic oscillator is x = A sin(ωt + φ), where x is the displacement from equilibrium, A is the amplitude, ω is the angular frequency, and φ is the phase angle. This equation can be applied to various systems to describe their motion, including mass-spring systems, pendulums, and electrical circuits.

4. How does the mass affect the motion of a simple harmonic oscillator?

The mass of a simple harmonic oscillator affects its motion by changing the period and frequency of oscillation. A larger mass will result in a longer period and lower frequency, while a smaller mass will lead to a shorter period and higher frequency. However, the amplitude and shape of the oscillation will remain the same regardless of the mass.

5. What are some real-life examples of Harmonic Oscillator equivalence?

Some real-life examples of Harmonic Oscillator equivalence include the motion of a swing, the vibrations of a guitar string, and the motion of a clock pendulum. All of these systems can be described by the same mathematical equation for a simple harmonic oscillator, demonstrating the principle of Harmonic Oscillator equivalence.

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