Ground State and Excited State Wave Function - Explained

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Discussion Overview

The discussion revolves around the differences between ground state and excited state wave functions in quantum mechanics, specifically in the context of the Schrödinger equation and the one-dimensional harmonic oscillator. Participants explore the characteristics of these wave functions, their mathematical properties, and their implications for further studies in quantum field theory.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants assert that ground state and excited state wave functions are different solutions to the Schrödinger equation.
  • It is noted that the exact shape of the wave functions depends on the potential involved, with the one-dimensional harmonic oscillator being a common example.
  • One participant questions whether the maximum value of the wave function for the ground state (n=0) is the same as that for the first excited state (n=1), suggesting a need for numerical evaluation.
  • Another participant emphasizes the importance of the harmonic oscillator in understanding concepts relevant to quantum field theory, including the number operator and creation/annihilation operators.
  • It is mentioned that the ground state wave function has no nodes, while excited states have nodes, with higher excited states exhibiting more nodes and undulations.

Areas of Agreement / Disagreement

Participants generally agree that ground and excited state wave functions are different, but there is no consensus on specific numerical comparisons or interpretations of their properties. The discussion includes multiple viewpoints on the implications for quantum field theory.

Contextual Notes

Some participants reference external resources for further understanding, but there are unresolved questions regarding the specific characteristics of the wave functions and their numerical evaluations.

Who May Find This Useful

This discussion may be of interest to students and enthusiasts of quantum mechanics, particularly those exploring wave functions, the Schrödinger equation, and quantum field theory concepts.

terp.asessed
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I am curious, if I were to draw a wave function, would one for ground state and one for excited sate be different? If different, could someone explain how and why? If you could, thanks!
 
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Well in general you can't draw wavefunctions because they are complex valued and are defined at points of three dimensional space.

But yes - the wave-functions are different.

The why is trivial - they are different solutions of the Schroedinger equation.

Thanks
Bill
 
They will be different, because they're different solutions to the same differential equation (the Schrödinger equation). If they weren't different they'd be the same solution so would describe the same state and be the same.

The exact shape of the wave functions will depend on the potential for a particular situation, but a good example is the one-dimensional harmonic oscillator. Google will find you a bunch of images of the wave function for various energy levels in that potential.
 
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Okay...so I am curious if the wave function (n=0) has the same maximum value as wave function (n=1)? I mean, based on the gif.
 
Awesome! I get it!
 
terp.asessed said:
Okay, thank you for the reply! Btw, I did google online, "one-dimensional harmonic oscillator" but am not sure which is exactly one that describes wave functions for various energy levels in that potential...is it this one?: http://upload.wikimedia.org/wikipedia/commons/e/e0/StationaryStatesAnimation.gif

Yes.

But I also want to mention this is a very important example for future studies into Quantum Field theory. You get to grips with things like the number operator, annihilation, and creation operators.

The reason its so important is if you do a Fourier transform on a quantum field each of the 'parts' of the transform act mathematically exactly the same as the harmonic oscillator - which is hardly surprising since, classically, you have gone to the frequency domain and the harmonic oscillator oscillates at a fixed frequency. You transform back and low and behold you see a quantum field consists of creation and annihilation operators that behave exactly the same as creating and annihilating particles of a certain momentum. Really deep and interesting stuff.

If that has whetted your appetite for QFT then I highly recommend the following:
https://www.amazon.com/dp/019969933X/?tag=pfamazon01-20

I am going through it right now.

You can tackle it after a basic course on QM.

Thanks
Bill
 
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Very simply, the ground state wavefunction won't have nodes. An excited state will have nodes, with higher excited levels having more nodes and more undulations. The spatial frequency of the wave will be higher.
 

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