What does a zero-value of the Born-condition mean?

[SeM]

Hi, I was wondering what is the physical meaning of these integrals:

\begin{equation}
\int_{-\infty}^{\infty} \psi \psi^* dx = 0
\end{equation}


\begin{equation}
\int_{-\infty}^{\infty} \psi x \psi^* dx = 0
\end{equation}

\begin{equation}
\int_{-\infty}^{\infty} \psi p \psi^* dx = NA
\end{equation}

?

 

[Demystifier]

Where did you find this expression?

 

[SeM]

Where did you find this expression?

By a function I have been calculating on

 

[Demystifier]

What is $t$? Time? 3-volume?

 

[SeM]

What is $t$? Time? 3-volume?

Sorry, I should have used x. Corrected.

 

[Demystifier]

(1) implies $\psi=0$.
(2) means that average position is zero.
(3) means that the average momentum is not well defined.

 

[SeM]

(1) implies $\psi=0$.

This part is not clear to me. Evidently, Psi and Psi* are hermitian counterparts, so their multiplication with one another ends up with zero.

(2) means that average position is zero.

This is OK. But does that mean the wavefunction has not physical application?

(3) means that the average momentum is not well defined.

This is fine. Does normalization solve this?

 

[Demystifier]

Evidently, Psi and Psi* are hermitian counterparts, so their multiplication with one another ends up with zero.

No. For properly normalized wave functions you should have $\int \psi^*\psi dx=1$.

But does that mean the wavefunction has not physical application?

No, such wave functions have a lot of applications. For instance, the ground state of harmonic oscillator has zero average position.

Does normalization solve this?

Probably not. Consider, for example, the delta-function $\delta(x)$. It has undefined average momentum and change of normalization does not help.

 

[SeM]

No. For properly normalized wave functions you should have $\int \psi^*\psi dx=1$.

Is there any chance of solving this constant by setting the integral:

\begin{equation}
N^2 \int_0^L \psi \psi^* dx = 1
\end{equation}

and solve for N?

I tried and got complex conjugates, in fact, not only one, but two.

No, such wave functions have a lot of applications. For instance, the ground state of harmonic oscillator has zero average position.

Thanks for this interesting point! Do you know of a paper that reports such a wavefunction?

Probably not. Consider, for example, the delta-function $\delta(x)$. It has undefined average momentum and change of normalization does not help.

What does one do with such wavefunctions without a well-defined momentum and kinetic energy?

 

[Demystifier]

Is there any chance of solving this constant by setting the integral:

\begin{equation}
N^2 \int_0^L \psi \psi^* dx = 1
\end{equation}

and solve for N?

Yes, provided that N is assumed to be real. (Otherwise, you should write $N^*N$ instead of $N^2$.)

I tried and got complex conjugates, in fact, not only one, but two.

Take into account that N is assumed to be real.

Thanks for this interesting point! Do you know of a paper that reports such a wavefunction?

Any textbook on quantum mechanics (QM) should do. Which book on QM do you use?

What does one do with such wavefunctions without a well-defined momentum and kinetic energy?

Such a $\delta$-function is an idealization of a more realistic case of a very narrow Gaussian. Even though the idealization is not realistic, it is OK as long as you only consider positions and not momenta or energy.

 

[SeM]

Yes, provided that N is assumed to be real. (Otherwise, you should write $N^*N$ instead of $N^2$.)

Take into account that N is assumed to be real.

Tried and got two weird values.

Let me share this strange wavefunction with the forum here:

\begin{equation}
\psi(x)= \frac{e^{-x(k_2-k_1)}\big(\sqrt{E-2\gamma^2}+\gamma i\big)}{2\sqrt{E-2\gamma^2}}-\frac{e^{-x(k_1+k_2)}\big(-\sqrt{E-2\gamma^2}+\gamma i\big)}{2\sqrt{E-2\gamma^2}}
\end{equation}

Any textbook on quantum mechanics (QM) should do. Which book on QM do you use?

Molecular Quantum Mechanics, there they normalize it, and get nice eigenfunctions. This function however, gave two horrible Normalization constants, of the length of half a page, and cannot really give a worse starting point for generating higher energy levels.

 

[Demystifier]

In your wave function I see only one (not two) implicit values of N, and I see nothing strange with it.

 

[SeM]

In your wave function I see only one (not two) implicit values of N, and I see nothing strange with it.

This function is without N. I had two initial conditions for the original ODE, first y(0)=1 and y'(0)=0 , and got this function. This function follows the integrals given in the first post.

Therefore I was looking for adding a normalization constant to the integrals.

 

[Demystifier]

This function follows the integrals given in the first post.

This function does not satisfy your Eq. (1).

 

[Demystifier]

Molecular Quantum Mechanics

Author please!

 

[SeM]

Author please!

Atkins and Friedman

 

[SeM]

This function does not satisfy your Eq. (1).

you need k_1 and k_2, these are:


\begin{equation}
k_1 = {\frac{\,\sqrt{E-2\,\gamma^2}}{\hbar}}
\end{equation}

\begin{equation}
k_2 = \frac{\gamma i}{\hbar}
\end{equation}


$\gamma = 5^{-28}$ and E is the zero point energy. If you still get that integral 1 is not satisfied, then MATLAB is crap

 

[Demystifier]

Atkins and Friedman

Are you a chemist? I ask because it looks like you miss the basic foundations of QM. In any case, study first Chapters 1 and 2 of the book. For instance, in Fig. 2.27 (fourth edition) all wave functions have zero average position.

 

[SeM]

Yes, but this model has not potential energy. and Yes I am a chemist.

What can I do to develop this model further as it does not follow the Hamiltonian on p 55, chapter 2, where fig 2.27 is.

Thanks!

 

[Demystifier]

p 55, chapter 2, where fig 2.27 is.

We must be using different editions of the book. Mine is the 4th edition. In any case, try to compute $\int \psi^*\psi dx$ again and show that it is a positive real number. You cannot make any progress before you do that.

 

[blue_leaf77]

Sorry to barge in, but your wavefunction model is proportional to $\cosh (k_1x)$, which is not square integrable for $x\in (-\infty,\infty)$. Moreover, the wavefunction is also proportional to an oscillatory function whose wave number is extremely large $~10^{-28}/10^{-34} \approx 10^6$. It can be the reason why MATLAB returns zero. Have you tried executing "format long" command to extend the decimal places in the output ?

 

[Demystifier]

Sorry to barge in, but your wavefunction model is proportional to $\cosh (k_1x)$, which is not square integrable for $x\in (-\infty,\infty)$. Moreover, the wavefunction is also proportional to an oscillatory function whose wave number is extremely large $~10^{-28}/10^{-34} \approx 10^6$. It can be the reason why MATLAB returns zero. Have you tried executing "format long" command to extend the decimal places in the output ?

His wave function probably needs to be defined in a box of finite length $L$, as suggested in post #9.

 

[Demystifier]

$\gamma = 5^{-28}$ ... If you still get that integral 1 is not satisfied, then MATLAB is crap

One of the first rules in any numerical computation (with MATLAB or anything else) is to first introduce new dimensionless variables (which are certain ratios of the dimensional ones) such that all relevant parameters are of the order of unity. Very big and very small numbers are always potentially dangerous in numerical computations.

 

[SeM]

We must be using different editions of the book. Mine is the 4th edition. In any case, try to compute $\int \psi^*\psi dx$ again and show that it is a positive real number. You cannot make any progress before you do that.

Thanks! Can you possibly scan the page or mention any of the subttiles of the page or preceeding pages, so I may look for it in the book?

I will recompute the integral tomorrow and see if it gives another value in the meantime.

Cheers

 

[SeM]

His wave function probably needs to be defined in a box of finite length $L$, as suggested in post #9.

Defining it in a box of length L, implies that I do this:


Psi(x) = 0 , L=0

Psi(x) = 0 , L=L


and I add a normalization constant N^2 to the Born intergral, and then use L=L and L=0 as the limits?

I then solve for this integral, but x will still be there...?

 

[Demystifier]

Thanks! Can you possibly scan the page or mention any of the subttiles of the page or preceeding pages, so I may look for it in the book?

Search for the section The harmonic oscillator - The solutions.

 

[Demystifier]

Defining it in a box of length L, implies that I do this:


Psi(x) = 0 , L=0

Psi(x) = 0 , L=L


and I add a normalization constant N^2 to the Born intergral, and then use L=L and L=0 as the limits?

I then solve for this integral, but x will still be there...?

You mean x=0 and x=L. When you integrate over x from 0 to L, then x is no longer there.

 

[SeM]

You mean x=0 and x=L. When you integrate over x from 0 to L, then x is no longer there.

Indeed! Sorry for that mistake. I will try this tomorrow,

All the best

 

[SeM]

You mean x=0 and x=L. When you integrate over x from 0 to L, then x is no longer there.

Search for the section The harmonic oscillator - The solutions.

Dear Demystifier, I have gone through the Further information on chap. 2.2 Harmonic Oscillator, and it appears as that I have to solve the Schrödinger eqn using the annhalation and creation operators, instead of solving it numerically, as I did and got that strange wavefunction.

Thanks for your tip. This becomes an entirely new project, because it's currently huge.

Thanks!

 

[Demystifier]

Well, if you want to learn quantum chemistry, you cannot avoid learning more about harmonic oscillator.

 

[SeM]

Well, if you want to learn quantum chemistry, you cannot avoid learning more about harmonic oscillator.

I look forward to it.