Linear Harmonic Oscillator Quantum Mechanics

[says]

Homework Statement

A linear harmonic oscillator with frequency ω = h_bar / M is at time t = 0 in the state described by the wave-function:
Ψ(x,0) = C( 1 + √2x) e^-x2/2

Determine the values of energy which can be measured in this state.

I'm not really sure where to start this question and was wondering if someone could help me get the ball rolling.

Homework Equations

The Attempt at a Solution

 

[blue_leaf77]

The question asks you to first recognize the eigenfunctions of linear harmonic oscillator. Go look up for it either in your notes or online.

 

[says]

Is there a different between the harmonic oscillator and the linear harmonic oscillator? I have a pretty good understanding of the harmonic oscillator but I can't really find any literature to see if there's a difference between the two.

 

[blue_leaf77]

Linear harmonic oscillator is the one whose potential energy is given by $\frac{1}{2}m\omega^2x^2$.

 

[says]

Do I have to work with ladder operators for this problem as well?

We know the quantum system has energy, E, which means it is in an energy eigenstate, Ψ. We don't know for certain what energy the system has, but we know the probabilities for various states. Energy must be an eigenvalue of the energy operator, so we say the system is in a superposition of eigenstates:

Ψ = Σ c_iΨ_i

|c|² is the probability the system has energy E_i

 

[blue_leaf77]

Do I have to work with ladder operators for this problem as well?

You don't need to.

We know the quantum system has energy, E, which means it is in an energy eigenstate, Ψ. We don't know for certain what energy the system has, but we know the probabilities for various states. Energy must be an eigenvalue of the energy operator, so we say the system is in a superposition of eigenstates:

Ψ = Σ ciΨi

|c|2 is the probability the system has energy Ei

Yes, you are in the right track. The idea is to expand to given wavefunction into the eigenfunctions of harmonic oscillator. The expansion generally spans all possible eigenfunctions,.i.e. infinity. However, in the particular problem at hand, there should be only a few of them whose expansion coefficients are not vanishing. You task is to find the eigenfunctions which corresponds to these non-vanishing coefficients.

 

[says]

I can't seem to find any reference material on the linear harmonic oscillator. It's all just harmonic oscillator stuff.

 

[says]

Like I know how to derive E₁ = 1/2 H_barw and E₂ = 3/2 H_barw but I don't know if this is what the question is asking.

We are given the solution that at the state at time t = 0 is a superposition of energy eigenstates
Ψ(0) = 1/√2 (Ψ₀ + Ψ₁)

I don't understand how to get from the original definition of the wavefunction to that.

 

[blue_leaf77]

I can't seem to find any reference material on the linear harmonic oscillator. It's all just harmonic oscillator stuff.

Check in those references if the potential being discussed satisifes the form I wrote in post #4. Your aim is to find the expression for the first few eigenfunctions.

 

[says]

H_hat = p_hat² / 2 + 1/2 ω² x²

The position operator x_hat² is just = to x so I just wrote x above. I've also set m=1 to get rid of it in the equation and make it a little simpler for now.

H_hat = (-ih_bar d/dx)² + 1/2 ω² x²

set h_bar = 1 because it clutters up the equation.

so (-i_bar d/dx)² = -d²/dx²

H_hat = -1/2 d²/dx² + 1/2 ω² x²

H | Ψ > = -1/2 d² | Ψ > / dx² + 1/2 ω² x² | Ψ>

= E | Ψ>

| Ψ> = e^-ωx2/2

Differentiating this twice we get:

d² | ψ> / dx² = -ω | Ψ> + ω²x² | ψ>

H | ψ> = -1/2 ( -ω | ψ > + ω² x² | ψ>) + 1/2 ω² x² | ψ>)

H | ψ> = ω/2 | ψ> = E | ψ>

E = ω/2
(subbing h_bar back in
E=h_barω/2
E=hf

 

[says]

From here I know if we use the a and a⁺ operators we can find 3/2 h_bar/2, 5/2, etc...

I don't understand why the question has defined the wavefunction Ψ(x,0) = C( 1 + √2x) e^-x2/2. Could you explain this to me conceptually?

i.e. What is C?

 

[blue_leaf77]

H_hat = p_hat² / 2 + 1/2 ω² x²

The position operator x_hat² is just = to x so I just wrote x above. I've also set m=1 to get rid of it in the equation and make it a little simpler for now.

H_hat = (-ih_bar d/dx)² + 1/2 ω² x²

set h_bar = 1 because it clutters up the equation.

so (-i_bar d/dx)² = -d²/dx²

H_hat = -1/2 d²/dx² + 1/2 ω² x²

H | Ψ > = -1/2 d² | Ψ > / dx² + 1/2 ω² x² | Ψ>

= E | Ψ>

| Ψ> = e^-ωx2/2

Differentiating this twice we get:

d² | ψ> / dx² = -ω | Ψ> + ω²x² | ψ>

H | ψ> = -1/2 ( -ω | ψ > + ω² x² | ψ>) + 1/2 ω² x² | ψ>)

H | ψ> = ω/2 | ψ> = E | ψ>

E = ω/2
(subbing h_bar back in
E=h_barω/2
E=hf

As I said, you only need to know the functional forms of the first few eigenfunctions of (linear) harmonic oscillator, like those given in http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc5.html. Analyze the form of the asked wavefunction $\psi(x,0)$ an compare it to the form of eigenfunctions of the linear harmonic oscillator to find out which of the expansion coefficients, $c_n$, in the infinite series $\psi(x,0) = \sum_n c_n u_n(x)$, where $u_n(x)$ is the eigenfunction of harmonic oscillator, have non-zero value.

 

[says]

Just looking at them now! I can see a lot of similarities between the first 4 normalised wavefunctions and the function I have been given in the question.

 

[blue_leaf77]

Now then check which $c_n$'s for those first few eigenfunctions have nonzero value.

 

[says]

I'm still a bit new to the expansion coefficients. Still trying to understand it all.
.
I'm a bit lost at this point

 

[says]

I don't understand what alpha or y are in the normalised wavefunctions

 

[says]

or C_i -- it was in my text. I understand the two wave functions are a superposition of each other, but there is no definition in my text as to what C_i is, or any example.

 

[blue_leaf77]

I don't understand what alpha or y are in the normalised wavefunctions

How $\alpha$ and $y$ are defined is already stated in that page, read more thoroughly.

I understand the two wave functions are a superposition of each other, but there is no definition in my text as to what Ci is, or any example.

Which wavefunctions are superposition of each other?
If you don't find how to calculate $c_n$ in your text, this means either you haven't read through it scrupulously or the level of the text is too advanced to be considered as introductory level quantum mechanics.

 

[says]

What is C_n called? I could look it up!

 

[blue_leaf77]

What is C_n called? I could look it up!

Expansion coefficient, sometimes it is also referred to as projection coefficient.

 

[says]

I've read through a few different bits on expansion coefficient but I'm still really confused...

 

[blue_leaf77]

Are you familiar with Dirac notation? Something which looks like $| \ldots \rangle$.

 

[says]

Yes

 

[blue_leaf77]

In Dirac notation, the state $\psi(x,0)$ can be written as $|\psi(x,0)\rangle$ and its expansion will look like $|\psi(x,0) \rangle = \sum_n c_n |u_n\rangle$, where $|u_n\rangle$ are eigenstate of harmonic oscillator. Now these eigenfunctions are a set of orthonormal vectors, which means $\langle u_m|u_n\rangle = \delta_{mn}$, i.e. that inner product will vanish unless $m=n$. Now you have an infinite terms in $\sum_n c_n |u_n\rangle$, if you want to "filter out" just one particular coefficient, say $c_p$, out of that series using the orthonormality property, what you will do?

 

[says]

Sorry I'm still a bit confused. I don't understand what U_n, U_m, δ_mn, m or n mean...

 

[blue_leaf77]

$|u_n\rangle$ and $|u_m\rangle$ are two eigenstates of harmonics oscillator, with $m,n=0,1,2,\ldots$. $\delta_{mn}$ is the so-called kronecker Dirac delta symbol and it has the values
$$
\delta_{mn} = \left\{
\begin{array}{lr}
1 & : m=n\\
0 & : m\neq n
\end{array}
\right.
$$
I think you seriously need to review your textbook from the introductory chapter. It's strange that you could give a sufficient explanation for the meaning of state expansion in post #5 yet you don't know yet what those notations mean.

 

[says]

Where should I start? I really wish I had a text that started with first principles.

 

[blue_leaf77]

I recommend "introduction to quantum mechanics" by D. Griffith.

 

[says]

Thanks, i will have a read. Anything in particular I should look at?

 

[blue_leaf77]

Put emphasis on chapter 2 and 3.