Simple expectation value calculation

In summary, the conversation is about finding the expectation value of x using the Gaussian Distribution. The integral to be evaluated is ∫xp(x)dx on (-∞,∞). The solution should be x = a since the distribution is centered at x = a. Different methods are discussed, including setting p'(x) = 0 and solving for x, and using integration by parts. The final solution is found to be √(π/a)/(2a), as e-x2 tends to zero much faster than x tends to infinity.
  • #1
Syrus
214
0

Homework Statement



This problem comes from the second edition of Griffiths's, Introduction to Quantum Mechanics.

Given the Gaussian Distribution: p(x) = Aec(x-a)2

find <x>, that is, the expectation (or mean) value of x.

Clearly, to do this you evaluate the following integral: ∫xp(x)dx on (-∞,∞). I've tried performing this integration analytically, and it seems to be more bothersome than a rectal exam. The solution should be x = a since, by definition, p is centered at x = a (this is evident graphically as well). Is a qualitative solution sufficient here? Another method (which obviously does not work in general) is to set p'(x) = 0 and solve for x since we know the Gaussian distrubtion contains only one extremum. Any advice on the best means to proceed?

Homework Equations


The Attempt at a Solution

 
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  • #2
∫Axec(x-a)2.dx
subst x+a for x:
∫A(x+a)ecx2.dx = ∫Axecx2.dx + a∫Aecx2.dx
= (1/2)∫Aecx2.dx2 + a
 
  • #3
I tried this already. Wouldn't the differential be required to change also? That is, dx becomes d(x+a).
 
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  • #4
EDIT* sorry, sorry, the differential is dx'/dx = d/dx[x+a], so dx' = dx.
 
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  • #5
Now let's talk about <x2>, I seem to encounter an indeterminate form upon evaluating the integral at bounds +/- ∞ (see the integral of x2e-cx2)
 
  • #6
Syrus said:
Now let's talk about <x2>, I seem to encounter an indeterminate form upon evaluating the integral at bounds +/- ∞ (see the integral of x2e-cx2)
Same substitution gets rid the -a. ∫x2e-cx2.dx (with a suitable multiplier to get total prob of 1) is just var(X), a standard result. To work it out for yourself, use integration by parts.
 
  • #7
I obtain the integral in the attached photo. Obviously there is a problem evaluating it at +/- ∞ because of the second (subtracted) term, no?
 

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  • #8
The attachment is illegible for me. Much too small. Did you try integration by parts?
##\int_{-\infty}^{+\infty} x^2e^{-cx^2}.dx = [x \int^x ye^{-cy^2}.dy]_{-\infty}^{+\infty} - \int_{-\infty}^{+\infty} \left(\int^x ye^{-cy^2}.dy\right) dx ##
 
  • #9
I think you're misunderstanding me. Look here for the integral of interest (the one you're attempting to solve by parts): http://integral-table.com/ and you'll see the "problem" I'm encountering with evaluating the result at +/- infinity.
 
  • #10
Syrus said:
I think you're misunderstanding me. Look here for the integral of interest (the one you're attempting to solve by parts): http://integral-table.com/ and you'll see the "problem" I'm encountering with evaluating the result at +/- infinity.
Number (70)? erf(-∞) = -1, erf(+∞) = +1. The other term vanishes at both extremes, leaving √(π/a)/(2a)
 
  • #11
Why does the other term vanish at both extremes? Isn't 0*infinity an indeterminate form?
 
  • #12
e-x2 tends to zero much faster than x tends to infinity.
 
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  • #13
There's an x multiplied by that too however, i.e. xecx2, so at each extreme it becomes +/- infinity * e^-infinity
 
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Related to Simple expectation value calculation

1. What is a simple expectation value calculation?

A simple expectation value calculation is a mathematical technique used in quantum mechanics to determine the expected value of a physical quantity, such as position or momentum, in a given quantum state. It involves taking the average of all possible outcomes of a measurement, weighted by their probability of occurrence.

2. How is a simple expectation value calculated?

To calculate a simple expectation value, one first needs to determine the wavefunction of the quantum system in question. This wavefunction is then used to calculate the probability of each possible measurement outcome. The expectation value is then found by multiplying each measurement outcome by its corresponding probability and summing all the results.

3. Can a simple expectation value calculation be used for any physical quantity?

Yes, a simple expectation value calculation can be used for any physical quantity that can be described by a quantum wavefunction. This includes properties such as energy, spin, and angular momentum.

4. Are there any limitations to using a simple expectation value calculation?

While a simple expectation value calculation is a useful tool in quantum mechanics, it does have some limitations. It can only provide an average value for a physical quantity and does not give information about the specific outcome of a measurement. Additionally, it may not accurately represent the behavior of a quantum system in certain situations, such as when the system is in a state of superposition.

5. How is a simple expectation value calculation used in experimental research?

In experimental research, a simple expectation value calculation is often used to predict the outcome of a measurement before it is performed. This can help researchers design experiments and interpret their results. It can also be used to compare the predicted values with the actual measured values, providing insight into the behavior of the quantum system being studied.

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