Ground State and Excited State Wave Function - Explained

[terp.asessed]

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!

 

[bhobba]

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

 

[Nugatory]

They will be different, because they're different solutions to the same differential equation (the Schrodinger 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.

 

[terp.asessed]

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

 

[Nugatory]

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, that's one.

 

[terp.asessed]

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.

 

[jtbell]

Look up the wave functions and plug in some numbers.

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc5.html

 

[terp.asessed]

Awesome! I get it!

 

[bhobba]

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

 

[Khashishi]

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.