Schrodinger Wave Equation question

Smythe

The attachment contains a Schrodinger wave equation question and the provided solution under it.

Though I got
d^2ψ/dx^2 = ψ(4y^2 x^2 - 2yx)

I'd be interested to learn if anyone gets this or the provided version.

Then this is substituted into SWE as per solution.

Then it states in the solution the x^0 and x^2 terms are equated. This I do not understand.

Where is the x^0 term? Though I realise this equals 1.

Also, why are the x^0 and terms x^2 equated? What does this achieve?

Thank you for any help!

Smythe

Also, why have they put together the 1st and 3rd terms together to equal zero, and then separately the 2nd and 4th terms to equal zero?

Why not the 1st with the 4th and the 2nd with the 3rd, for example?

And when the 2nd and 4th terms were put together to equal zero, why did the x^2 terms drop off?

dlouth15

Smythe wrote: »

Then it states in the solution the x^0 and x^2 terms are equated. This I do not understand.

Where is the x^0 term? Though I realise this equals 1.

Also, why are the x^0 and terms x^2 equated? What does this achieve?

I'll try and help with this bit.

If you are told, say, that [latex]ax^{2}+b\equiv7x^{2}+3[/latex], then we know that the only way for the identity to work is if [latex]a=7[/latex] and [latex]b=3[/latex]. This is called equating the coefficients, and here we've equated the [latex]x^2[/latex] coefficient, i.e. [latex]a[/latex] and the [latex]x^0[/latex] coefficient, i.e. [latex]b[/latex] with the values [latex]7[/latex] and [latex]3[/latex] respectively.

You can do something similar with the expression you have. Get all the terms involving [latex]\gamma[/latex] to one side and all the rest to the other side of the equation. Then we have

[latex]4\gamma^{2}x^{2}-2\gamma=\left(\frac{m\omega}{\hbar}\right)^{2}x^{2}-\frac{2mE}{\hbar}[/latex]

Then equate the [latex]x^2[/latex] term on the LHS with the one on the RHS and likewise the constant terms (i.e. the [latex]x^0[/latex] terms) on either side with each other.

Ciaran

Smythe wrote: »

The attachment contains a Schrodinger wave equation question and the provided solution under it.

Though I got
d^2ψ/dx^2 = ψ(4y^2 x^2 - 2yx)

I'd be interested to learn if anyone gets this or the provided version.

Then this is substituted into SWE as per solution.

Parts of the question didn't show up for me in the document so I can't help with this part.

Then it states in the solution the x^0 and x^2 terms are equated. This I do not understand.

Where is the x^0 term? Though I realise this equals 1.

The x^0 terms are the ones that don't contain x; -2y and 2mE/h^2 in this case.

Also, why are the x^0 and terms x^2 equated? What does this achieve?

The equation after "Substituting into SWE" has a left hand side equal to zero. This is true for all values of x so the parts that the parts that depend on x^2 must equal zero on their own and so must the parts that don't depend on x.

Smythe

Cheers guys, I actually understand this now.

The part which I thought was incorrect in the solution is actually correct. I had made a minor error in calculating this.

Thanks very much for the help!