Expectation of energy for a wave function

[thomas19981]

Homework Statement

At $t = 0$, a particle of mass m in the harmonic oscillator potential, $V(x) = \frac1 2 mw^2x^2$ has the wave function:$$\psi(x,0)=A(1-2\sqrt\frac{mw} {\hbar} x)^2e^{\frac{-mw}{2\hbar}x^2}$$

where A is a constant

If we make a measurement of the energy, what possible values might we obtain and what is the probability of obtaining each of these values. Hence determine the expectation value of the energy.

Homework Equations

$E_n=(n+\frac12)\hbar w$

The Attempt at a Solution

Firstly the first part of the question asked me to determine $A=\frac15(\frac{mw}{\hbar \pi})^\frac14$.

So I know to determine the the probability of measuring an energy $E_n$ I need to determine the the coefficient $c_n$ of each smaller wave function then square it. I also know that that $E_n=(n+\frac12)\hbar w$ and to find the expectation of the energy I would just sum over all n $E_nc_n^2$The only problem is that I don't know how to decompose the wave function given down so that I get $\psi(x,0)=c_1\psi_1(x)+c_2\psi_2(x)+ c_3\psi_3(x)+...$ . Once I know how to do this the rest will be easy.

Thank you in advance

 

[Ray Vickson]

Homework Statement

At $t = 0$, a particle of mass m in the harmonic oscillator potential, $V(x) = \frac1 2 mw^2x^2$ has the wave function:$$\psi(x,0)=A(1-2\sqrt\frac{mw} {\hbar} x)^2e^{\frac{-mw}{2\hbar}x^2}$$

where A is a constant

If we make a measurement of the energy, what possible values might we obtain and what is the probability of obtaining each of these values. Hence determine the expectation value of the energy.

Homework Equations

$E_n=(n+\frac12)\hbar w$

The Attempt at a Solution

Firstly the first part of the question asked me to determine $A=\frac15(\frac{mw}{\hbar \pi})^\frac14$.

So I know to determine the the probability of measuring an energy $E_n$ I need to determine the the coefficient $c_n$ of each smaller wave function then square it. I also know that that $E_n=(n+\frac12)\hbar w$ and to find the expectation of the energy I would just sum over all n $E_nc_n^2$The only problem is that I don't know how to decompose the wave function given down so that I get $\psi(x,0)=c_1\psi_1(x)+c_2\psi_2(x)+ c_3\psi_3(x)+...$ . Once I know how to do this the rest will be easy.

Thank you in advance

You need to expand $\psi(x)$ into something resembling a Fourier series, except that you must expand in terms of the eigenfunctions of the Harmonic Oscillator's hamiltonian rather than sines/cosines. Are you familiar with "orthogonal expansions" other than Fourier series? If not, Google is your friend.

 

[PeroK]

Homework Statement

At $t = 0$, a particle of mass m in the harmonic oscillator potential, $V(x) = \frac1 2 mw^2x^2$ has the wave function:$$\psi(x,0)=A(1-2\sqrt\frac{mw} {\hbar} x)^2e^{\frac{-mw}{2\hbar}x^2}$$

where A is a constant

If we make a measurement of the energy, what possible values might we obtain and what is the probability of obtaining each of these values. Hence determine the expectation value of the energy.

Homework Equations

$E_n=(n+\frac12)\hbar w$

The Attempt at a Solution

Firstly the first part of the question asked me to determine $A=\frac15(\frac{mw}{\hbar \pi})^\frac14$.

So I know to determine the the probability of measuring an energy $E_n$ I need to determine the the coefficient $c_n$ of each smaller wave function then square it. I also know that that $E_n=(n+\frac12)\hbar w$ and to find the expectation of the energy I would just sum over all n $E_nc_n^2$The only problem is that I don't know how to decompose the wave function given down so that I get $\psi(x,0)=c_1\psi_1(x)+c_2\psi_2(x)+ c_3\psi_3(x)+...$ . Once I know how to do this the rest will be easy.

Thank you in advance

Do you know the eigenstates of the harmonic oscillator?

 

[thomas19981]

Do you know the eigenstates of the harmonic oscillator?

Are the eigenstates the $\psi_n$? If it is then we have been given them in the lecture notes as $A_n(a_+)^n\psi_0(x)$ where $\psi_0=(\frac{mw}{\pi\hbar})^{0.25}e^{-\frac{mw}{2 \hbar}x^2}$ and $a_+$ is one of the ladder operators.

 

[PeroK]

Are the eigenstates the $\psi_n$? If it is then we have been given them in the lecture notes as $A_n(a_+)^n\psi_0(x)$ where $\psi_0=(\frac{mw}{\pi\hbar})^{0.25}e^{-\frac{mw}{2 \hbar}x^2}$ and $a_+$ is one of the ladder operators.

Yes, that's them. Can you see which eigenstates you'll need for your wavefunction?

 

[thomas19981]

Yes, that's them. Can you see which eigenstates you'll need for your wavefunction?

No but to determine them would I find $\psi_0$, $\psi_1$, $\psi_2$... then sum them till I find something resembling the wavefunction given in the question?

 

[PeroK]

No but to determine them would I find $\psi_0$, $\psi_1$, $\psi_2$... then sum them till I find something resembling the wavefunction given in the question?

Yes, you'll need to find the eigenstates. But, I would expect you to have reference to them for this question. I wouldn't expect that you have to derive them for this question. I may be wrong, but don't you have a list of them somewhere?

 

[thomas19981]

Yes, you'll need to find the eigenstates. But, I would expect you to have reference to them for this question. I wouldn't expect that you have to derive them for this question. I may be wrong, but don't you have a list of them somewhere?

I don't think so but how would I know at what n to stop cause in theory I would just keep applying $a_+$ and get infinitely many wave functions?

 

[PeroK]

I don't think so but how would I know at what n to stop cause in theory I would just keep applying $a_+$ and get infinitely many wave functions?

Do you not even have $\psi_0$? If you start deriving them, you should soon see when you have all you need. Hint: each function has an increasing polynomial in $x$ times the common exponential function. I assume the term "Hermite" polynomial is not familiar to you?

 

[TSny]

Firstly the first part of the question asked me to determine $A=\frac15(\frac{mw}{\hbar \pi})^\frac14$.

Looks good.

The only problem is that I don't know how to decompose the wave function given down so that I get $\psi(x,0)=c_1\psi_1(x)+c_2\psi_2(x)+ c_3\psi_3(x)+...$ . Once I know how to do this the rest will be easy.

Hello.
Try to express $(1-2\sqrt\frac{mw} {\hbar} x)^2$ as a linear combination of Hermite polynomials $H_n\left(\frac{m \omega}{\hbar}x \right)$. You should be able to do this "by inspection" if you have a list of the first few Hermite polynomials.

EDIT: Oh, I see I'm late in posting. It appears that you might not be familiar with the individual energy eigenfunctions expressed in terms of Hermite polynomials. If this is the case, I'm not sure what aspects of the harmonic oscillator energy eigenfunctions that you are familiar with.

 

[Ray Vickson]

I don't think so but how would I know at what n to stop cause in theory I would just keep applying $a_+$ and get infinitely many wave functions?

Yes, that might (or might not) happen. You really do need to know (or find, or look up) the eigenfunctions $\psi_n$; if your textbook does not list them, go to the library and look in another book. Alternatively, look on-line.

 

[thomas19981]

Yes, that might happen. You really do need to know (or find, or look up) the eigenfunctions $\psi_n$; if your textbook does not list them, go to the library and look in another book. Alternatively, look on-line.

Ok I think I get it. I would need to expand up to $\psi_2$ to get a term involving $x^2$ then I would normalise each of $\psi_0$, $\psi_1$, $\psi_2$ to get the value of $A_0,A_1,A_2$ respectively then compare the sum of these to the given wave function in the question to determine the numerical coefficients which would in turn give me the probability for each eigenstate. Is this idea correct?

 

[PeroK]

Ok I think I get it. I would need to expand up to $\psi_2$ to get a term involving $x^2$ then I would normalise each of $\psi_0$, $\psi_1$, $\psi_2$ to get the value of $A_0,A_1,A_2$ respectively then compare the sum of these to the given wave function in the question to determine the numerical coefficients which would in turn give me the probability for each eigenstate. Is this idea correct?

Yes, essentially. Once you have the eigenfunctions it's just a bit of linear algebra.

 

[thomas19981]

Yes, essentially. Once you have the eigenfunctions it's just a bit of linear algebra.

Ok thank you all !