The Energy Levels of a Quantum Harmonic Oscillator

[mrausum]

I've followed this:

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc3.html#c1,

up to the part where it gets to here:

.

The guide says: "Then setting the constant terms equal gives the energy"? Am I being stupid? I really can't see where that equations come from.

 

[kanato]

They have an equation of the form
$$a + b x^2 = c$$
which is true for all values of x. The only way for that to be true for all values of x is for a = c and b = 0. a and c are the constant terms they talk about, they do not depend on x. It gives the energy because the RHS of the equation only has the energy in it.

 

[mrausum]

They have an equation of the form
$$a + b x^2 = c$$
which is true for all values of x. The only way for that to be true for all values of x is for a = c and b = 0. a and c are the constant terms they talk about, they do not depend on x. It gives the energy because the RHS of the equation only has the energy in it.

Ah, of course. Thanks. Is it just by chance that the ground state energy is equal to the lowest possible value calculated from the uncertainty principle? I'm guessing the ground state energy for other potential functions is greater than that from the UP?

Also, is this:

just a particular integral that happens to give the lowest energy? I'm reading on now and it looks like the general solution is a ton of maths :(