Schrodinger equation problem : part 2

[noblegas]

Homework Statement

b)Use the equations <br /> a_{-}|\varphi_0\rangle=0 \implies \left(\left(\frac{k}{2}\right)^{1/2}x-\left(\frac{1}{2k/(2m)^(1/2)}\right)^{1/2}\frac{d}{d x}\right)\varphi_0(x)=0<br /> and <br /> a_{+}|\varphi_0\rangle=0 \implies \left(\left(\frac{k}{2}\right)^{1/2}x+\left(\frac{1}{2k/(2m)^(1/2)}\right)^{1/2}\frac{d}{d x}\right)\varphi_0(x)=0<br /> to express as a linear combination of the lowering and raising operators a_ and a+, and then use the operator methods to show that x<br /> \varphi_0(x)<br /> is the(not normalized) wave function for the first excited state;

c) Use the properties of the raising operator to argue that the wave function for the nth energy level, with E_n=(n+1/2)*omega*\hbar, for the oscillator is a polynomial multiplied by a gaussian exp(-c*x^2), and find the order of the polynomial

Homework Equations

The Attempt at a Solution

b) I don't think it is diffiicult to find the linear combination; c*(a_+ a_+)=c*2*(K/2)^(1/2); Not sure how I would show that it does not normalized for the first excited state but would I start the problem in this manner? a* <br /> \varphi_1(x)<br />=0, to find <br /> \varphi_1(x)<br /> and plug
<br /> <br /> \int_{-\infty}^{\infty} |\varphi_1(x)|^2 dx=1<br /> <br /> ; How would I not know that x<br /> \varphi_0(x)<br /> is not normalizable?

c) <br /> a_{+}|\varphi_0\rangle=0 \implies \left(\left(\frac{k}{2}\right)^{1/2}x+\left(\frac{1}{2k/(2m)^(1/2)}\right)^{1/2}\frac{d}{d x}\right)\varphi_0(x)=0<br />
Not sure what they are asking for.

 

[maverick280857]

The problem asks you to analyze the ground state of the harmonic oscillator wavefunction without explicitly getting into special functions. (The form of the ground state wavefunction is rather simple.)

You have been given the action of the raising and lowering operators in the position basis, which turn out to be differential equations of constraint on the ground state. But while I understand that

a_{-}|\psi_{0}\rangle = 0

as this is the defining equation for the ground state, I don't see how a_{+}|\psi_{0}\rangle = 0. In fact, the left hand side equals (give or take a constant factor) the ket |\psi_{1}\rangle.

[Edit: Can you clean up your latex a bit, esp for your attempted solution of part (c).]

 

[gabbagabbahey]

Your problem statement for b) is not very clear. What exactly are you being asked to express as a linear combination of raising and lowering operators? The X operator? The P operator? Something else entirely?

Also, you are using an incorrect expression for the raising and lowering operators in the x-basis...why is there a (2m)^{1/2} in your expression?

 

[noblegas]

The problem asks you to analyze the ground state of the harmonic oscillator wavefunction without explicitly getting into special functions. (The form of the ground state wavefunction is rather simple.)

You have been given the action of the raising and lowering operators in the position basis, which turn out to be differential equations of constraint on the ground state. But while I understand that

a_{-}|\psi_{0}\rangle = 0

as this is the defining equation for the ground state, I don't see how a_{+}|\psi_{0}\rangle = 0. In fact, the left hand side equals (give or take a constant factor) the ket |\psi_{1}\rangle.

[Edit: Can you clean up your latex a bit, esp for your attempted solution of part (c).]

<br /> <br /> a_{-}|\varphi_0\rangle=0 \implies \left(\left(\frac{k}{2}\right)^{1/2}x-\left(\frac{1}{2/(\hbar/(2m))}\right)^{1/2}\frac{d}{d x}\right)\varphi_0(x)=0<br /> <br />

<br /> <br /> a_{+}|\varphi_0\rangle=0 \implies \left(\left(\frac{k}{2}\right)^{1/2}x+\left(\frac{1}{2(\hbar(2m)/)}\right)^{1/2}\frac{d}{d x}\right)\varphi_0(x)=0<br /> <br />

plancks constant, K parameter constant and m should all be within the operator equations; I am still trying to get acquainted with using latex; Planck constant should be the numerator btw; These are the equations I am told to work with; Now that I have phi_0 from part a, I need to used the expression \varphi_0*x=0 to show that the expression does not normalized; How would I know that the expression does not normalized?

Should I express x in the terms operator a_-

 

[gabbagabbahey]

It seems to me that you are supposed to first express the operator X in terms of the raising and lowering operators, then use that to show that X|\varphi_0\rangle\propto|\varphi_1\rangle

 

[noblegas]

It seems to me that you are supposed to first express the operator X in terms of the raising and lowering operators, then use that to show that X|\varphi_0\rangle\propto|\varphi_1\rangle

Would I integrate x using the operators;


<br /> <br /> <br /> a_{+}|\varphi_0\rangle=0 \implies \left(\left(\frac{k}{2}\right)^{1/2}x+\left(\frac{1}{2(\hbar(2m)/)}\right)^{1/2}\frac{d}{d x}\right)\varphi_0(x)=0<br /> <br /> <br /> and
<br /> <br /> <br /> a_{-}|\varphi_0\rangle=0 \implies \left(\left(\frac{k}{2}\right)^{1/2}x-\left(\frac{1}{2/(\hbar/(2m))}\right)^{1/2}\frac{d}{d x}\right)\varphi_0(x)=0<br /> <br /> <br />; Sorry for all the questions; I am working all by myself and usually don't join study groups;

 

[gabbagabbahey]

What do you mean 'integrate x'?

You have an expression for the raising an lowering operators,

a_{+}=\left(\frac{k}{2}\right)^{1/2}x+\frac{\hbar}{(2m)^{1/2}}\frac{d}{dx}

a_{-}=\left(\frac{k}{2}\right)^{1/2}x-\frac{\hbar}{(2m)^{1/2}}\frac{d}{dx}

So what is the operator x in terms of those two raising and lowering operators?

What does that make x\varphi_0?

 

[noblegas]

What do you mean 'integrate x'?

You have an expression for the raising an lowering operators,

a_{+}=\left(\frac{k}{2}\right)^{1/2}x+\frac{\hbar}{(2m)^{1/2}}\frac{d}{dx}

a_{-}=\left(\frac{k}{2}\right)^{1/2}x-\frac{\hbar}{(2m)^{1/2}}\frac{d}{dx}

So what is the operator x in terms of those two raising and lowering operators?

What does that make x\varphi_0?

a_{+}+\frac{\hbar}{(2m)^{1/2}}\frac{d}{dx}=\left(\frac{k}{2}\right)^{1/2}x

a_{-}-\frac{\hbar}{(2m)^{1/2}}\frac{d}{dx}=\left(\frac{k}{2}\right)^{1/2}x,

x\varphi_0 would be from part a of the problem correct?

I ended up with this expression; x=(2/k)^(1/2)*(a_-+a_+)

 

[gabbagabbahey]

a_{+}+\frac{\hbar}{(2m)^{1/2}}\frac{d}{dx}=\left(\frac{k}{2}\right)^{1/2}x

a_{-}-\frac{\hbar}{(2m)^{1/2}}\frac{d}{dx}=\left(\frac{k}{2}\right)^{1/2}x

That's not what I meant.

You have two equations, involving the operators 'x', 'd/dx' and the raising and lowering operators. You should be able to combine your two equations to eliminate 'd/dx' and leave you with a single equation that involves only 'x' and the raising and lowering operators. You can then solve that equation for 'x'.

x\varphi_0 would be from part a of the problem correct?

No, in part (a) you found \varphi_0(x), but you don't need that result for part (b)

 

[noblegas]

That's not what I meant.

You have two equations, involving the operators 'x', 'd/dx' and the raising and lowering operators. You should be able to combine your two equations to eliminate 'd/dx' and leave you with a single equation that involves only 'x' and the raising and lowering operators. You can then solve that equation for 'x'.



No, in part (a) you found \varphi_0(x), but you don't need that result for part (b)

did you see my recently edited post

 

[noblegas]

I get it now; x <br /> \varphi_0<br />x=<br /> \varphi_0<br />(2/k)^(1/2)*(a_-+a_+) and I would integrate this expression to obtain a new expression for <br /> \varphi_0<br /> I don't think this expression is normalizable because everytime I attempt to integrate it <br /> \varphi_0<br /> cancels itself out

 

[gabbagabbahey]

did you see my recently edited post

I did now.

I ended up with this expression; x=(2/k)^(1/2)*(a_-+a_+)

Yes.

and I would integrate this expression to obtain a new expression for <br /> \varphi_0<br />

No. How would integrating the expression for the operator 'x' give you an expression for \varphi_0?!

Instead just multiply your expression for x by \varphi_0:

x\varphi_0=\left(\frac{2}{k}\right)^{1/2}\left(a_-+a_+\right)\varphi_0=\left(\frac{2}{k}\right)^{1/2}a_{-}\varphi_0+\left(\frac{2}{k}\right)^{1/2}a_{+}\varphi_0

But from the definition of the lowering operator, a_{-}\varphi_0=____? And from thew definition of the raising operator, a_{+}\varphi_0=____?

 

[noblegas]

I did now.



Yes.



No. How would integrating the expression for the operator 'x' give you an expression for \varphi_0?!

Instead just multiply your expression for x by \varphi_0:

x\varphi_0=\left(\frac{2}{k}\right)^{1/2}\left(a_-+a_+\right)\varphi_0=\left(\frac{2}{k}\right)^{1/2}a_{-}\varphi_0+\left(\frac{2}{k}\right)^{1/2}a_{+}\varphi_0

But from the definition of the lowering operator, a_{-}\varphi_0=____? And from thew definition of the raising operator, a_{+}\varphi_0=____?

I know from the definition for the lower operator , \varphi_0*a_=0 and therefore x =(2/k)^(1/2) a_+. Now I would apply E_n=(.5+n)

ground state E_0=.5*omega*\hbar
first excited state E_1=3/2*omega*\hbar

 

[gabbagabbahey]

I know from the definition for the lower operator , \varphi_0*a_=0

Yes.

and therefore x =(2/k)^(1/2) a_+.

No.

x=(2/k)^(1/2) (a_+ + a_-)

Just because the effect of acting a_- on a certain state produces a result of zero does not mean that a_- is zero.

You can however say that x\varphi_0=\left(\frac{2}{k}\right)^{1/2}a_+ \varphi_0

Now I would apply E_n=(.5+n)

Not quite, this is not about energy eigenvalues. This is about the effect of the raising operator on a given eigenstate...when the raising operator acts on an eigenstate, it transforms it into another eigenstate with a higher energy. By definition, \varphi_0 is the lowest energy eigenstate, so the eigenstate with an energy level of one quanta higher is by definition, \varphi_1.

Therefor a_+\varphi_0=\varphi_1...follow?

So x\varphi_0=____?

 

[noblegas]

Yes.



No.

x=(2/k)^(1/2) (a_+ + a_-)

Just because the effect of acting a_- on a certain state produces a result of zero does not mean that a_- is zero.

You can however say that x\varphi_0=\left(\frac{2}{k}\right)^{1/2}a_+ \varphi_0


Not quite, this is not about energy eigenvalues. This is about the effect of the raising operator on a given eigenstate...when the raising operator acts on an eigenstate, it transforms it into another eigenstate with a higher energy. By definition, \varphi_0 is the lowest energy eigenstate, so the eigenstate with an energy level of one quanta higher is by definition, \varphi_1.

Therefor a_+\varphi_0=\varphi_1...follow?

So x\varphi_0=____?

x\varphi_0=\left(\frac{2}{k}\right)^{1/2}\varphi_1

 

[gabbagabbahey]

Right, which is a non-normalized version of the 1st excited state. (\varphi_1 is presumably normalized, so \left(\frac{2}{k}\right)^{1/2}\varphi_1 is presumably unnormalized)

 

[noblegas]

in part c) E_n=(n+1/2)*omega <br /> \hbar<br /> should I transformed this expression to the gaussian expression?

Would this equation be relevant: H(a*\varphi )=(E-\hbar *omega)(a*\varphi )

 

[noblegas]

err... bump! just like in part b, do I need to show that x* <br /> \varphi_n<br />=(2/K)^(1/2)*<br /> \varphi_(n+1)<br />

1 is supposed to be added to the subscript along with n

 

[gabbagabbahey]

err... bump! just like in part b, do I need to show that x* <br /> \varphi_n<br />=(2/K)^(1/2)*<br /> \varphi_(n+1)<br />

1 is supposed to be added to the subscript along with n

Not quite, since a_{-}\varphi_{n-1} is only equal to zero if n=1...what do you get when you apply your expression for the x-operator to \varphi_{n-1}?...can you use that, along with the fact that in part (a) you found that \varphi_0(x) was Gaussian, to argue that \varphi_n=P(x)e^{-cx^2} for some polynomial P(x)?

 

[noblegas]

Not quite, since a_{-}\varphi_{n-1} is only equal to zero if n=1...what do you get when you apply your expression for the x-operator to \varphi_{n-1}?...can you use that, along with the fact that in part (a) you found that \varphi_0(x) was Gaussian, to argue that \varphi_n=P(x)e^{-cx^2} for some polynomial P(x)?

x*\varphi_{n-1}= (2/K)^(1/2)*\varphi_(n),since
<br /> \varphi_n=P(x)e^{-cx^2}<br />, then x*\varphi_{n-1}= (2/K)^(1/2)*\varphi_(n)= x*\varphi_{n-1}= (2/K)^(1/2)*<br /> \varphi_n=P(x)e^{-cx^2}<br />, then x*\varphi_{n-1}=(2/K)^(1/2)*<br /> \varphi_n=P(x)e^{-cx^2}<br />

 

[gabbagabbahey]

x*\varphi_{n-1}= (2/K)^(1/2)*\varphi_(n),since
<br /> \varphi_n=P(x)e^{-cx^2}<br />, then x*\varphi_{n-1}= (2/K)^(1/2)*\varphi_(n)= x*\varphi_{n-1}= (2/K)^(1/2)*<br /> \varphi_n=P(x)e^{-cx^2}<br />, then x*\varphi_{n-1}=(2/K)^(1/2)*<br /> \varphi_n=P(x)e^{-cx^2}<br />

You seem to be going in circles, \varphi_n=P(x)e^{-cx^2} is what you are asked to prove, it should not be one of the premises of your proof!

Also, re-read the 1st part of my last post...

x\varphi_{n-1}\neq \left(\frac{2}{K}\right)^{1/2}\varphi_n

for arbitrary values of 'n'...what do you actually get when you apply the expression for 'x' that you found in part (b)?