Examples of common quantum mechanical procedures and calculations carried out in Mathcad.

Transcription

1 Eample_QM_calculations.mcd page Eamples of common quantum mechanical procedures and calculations carried out in Mathcad. Erica Harvey Fairmont State College Department of Chemistry Fairmont State University Fairmont, WV 6554 Content: Given a function, use Mathcad to normalize the function, make plots of the probability amplitude and probability density versus position, decide whether or not the function is an eigenfunction of a specified operator and evaluate the eigenvalue if possible, evaluate whether or not the function is orthogonal to another function, calculate an epectation value for some observable using the function, calculate an uncertainty (standard deviation and variance) in the epectation value and calculate the probability of finding a particle described by the function in a specified region. y = e α Given this unnormalized wavefunction that describes some particle, answer the questions below. Assume: is greater than -infinity; is less than +infinity; in other words the "space" for this particle includes everywhere on the -ais.. Normalize the function. What is N? α := 3 Alpha is a constant, 3 was simply chosen for quick comparison purposes. y := e α ψ = N o y Define a wavefunction that includes the function above and a constant that will ensure the function is normalized. The constant (called a normalization constant) is given the symbol N.o to avoid redefining the Newton unit in Mathcad. The wavefunction, Psi() is also called the probability amplitude, where is the position variable. ψ star = N o y This function, psi-star(), would be the comple conjugate, however the orginal function does not contain an imaginary (i) term so it is

2 Eample_QM_calculations.mcd page just the same as Ψ(). ψ star ψ = This is a general way to write the normalization condition for a wavefunction describing a particle that eists somewhere in all of one-dimensional space. lower := upper := Specify the upper and lower limits of the entire region where the particle eists. These limits can be different for different physical systems and depend on the model you are using. upper lower ( N o y ) = This is the same normalization condition as above, with the function and normalization constant substituted in for the wavefunction and its comple conjugate. The limits of the integration have also been specified in an easier-to-modify way. N o := If I pull the normalization constant out of the integral and solve for it, I get the epression at the left. upper y lower N o = This is the value of the normalization constant. To make sure this is the correct normalization constant, I can plug the whole wavefunction back into the normalization conditon and make sure it is true. ψ := N o y I redefined the functions to include the N.o value I just got. These definitions are used in most of the rest of the document. ψ star := N o y upper ψ star ψ = lower upper lower ψ star ψ When I type the regular equals sign OR the control-period equals sign after the integral, it shows the value is indeed, as epected for a normalized wavefunction. This is the probability of finding a particle in the region between.lower and.upper, where.lower and.upper define the edges of the entire region where the particle eists.

3 Eample_QM_calculations.mcd page 3. Make a nicely-scaled and labeled plot of the probability amplitude versus position probability amplitude vs. position ψ 0.5 where: y-ais = ψ(); the probability amplitude -ais = ; the position Make a nicely-scaled and labeled plot of the probability density versus position probability density vs. position probability density ψ 0.5 Note that the probability density is sharper than the probability amplitude position If we did have an imaginary component in our wavefunction, we would have to be more careful and use the following, more precise, definition for probability density. This definition simplifies to psi-squared when the wavefunction isn't imaginary. pr density := ψ star ψ

4 Eample_QM_calculations.mcd page 4 probability density vs. position probability density pr density 0.5 For this wavefunction, the two plots are identical position 4. Calculate the probability of finding a particle described by the function in a specified region that can be easily changed. For comparison, submit your document with = 0 and = 0.5. := 0 Specify the limits of the region here. := 0.5 ψ star ψ = 0.39 This is the probability of finding a particle in the region between and. 5. Calculate the epectation value for position a average = ψ A operator ψ The epectation value (a.average or <a> in the case at left) for an observable or property is calculated by finding the operator that describes that quantity (A.operator in the generic case at left), finding a normalized wavefunction that describes the system under consideration, and using the integral definition for epectation value, shown at the left.

5 Eample_QM_calculations.mcd page 5 ave := ave = 0 ψ star ψ I called the epectation value for the.ave because that is really what an epectation value tells you - the average epected when many measurments have been taken. Also Mathcad isn't happy about the symbol commonly used for epectation value: <>. Note that the limits of integration are set to infinity and negative infinity because this particle eists over that entire range. The epectation value for position of the particle described by the wavefunction in this problem is just 0! So that could mean that half the time it is found at -values above 0 and half the time it is found at -values below 0. It might also mean that it is always found at 0. We can't tell the difference just by reporting an average. 6. Calculate an uncertainty (standard deviation and variance) in the epectation value for position Variance in a quantity can be calculated simply as the epectation value for the quantity squared (position squared in this case, < >) minus the (epectation value for the quantity)-squared (<> in this case.) σ squared := ψ star ψ ψ star ψ this is the formula for variance σ squared = 0.67 σ := σ squared this is the standard deviation σ = 0.408

6 Eample_QM_calculations.mcd page 6 7. Decide whether or not the normalized function is an eigenfunction of the one-dimensional kinetic energy operator and evaluate the associated eigenvalue if possible. h bar d T Op = m o this is the kinetic energy operator for a single particle (one dimension) Note that there is a black bo that shows the operator is waiting to operate on something. T Op Ψ = k Ψ This equation would have to be satisfied in order for the original function to be an eigenfunction of the kinetic energy operator. Here, k is some constant that we could find by operating on the function (if it turns out to be an eigenfunction.) Ψ 0 := N norm e α 0 I redefined the normalized wavefunction here using all new variable symbols because it will let me keep them as symbols when I do the symbolic manipulations in the net steps. (Since N.0 and α are defined with numbers above, if I kept them in place Mathcad would substitute those numbers in below and obscure the meaning of the differentials. Try it yourself to see the difference.) In the step below I literally operate on the wavefunction with the T.Op kinetic energy operator and the symbolic (control-period) equals sign. This makes Mathcad take the second derivative of the wavefunction and multiply it by the constants specified in T.Op. h bar d m o ( Ψ 0 ) h bar m o α 0 e N norm α 0 + N norm α 0 e α 0 h bar m o α 0 e N norm α 0 + N norm α 0 e α 0 factor this result to separate: *constants *orginial function *any other variables If this wavefunction is an eigenfunction of the kinetic energy operator, I should be able to factor out the original wavefunction and have the rest of the stuff be simply a constant. If it isn't simply a constant (or collection of constants) then the function isn't an eigenfunction of this particular operator.

7 Eample_QM_calculations.mcd page 7 these are the two terms after factoring the equation above by hand: h bar m o α 0 + α 0 α 0. N norm e constants EXCEPT for the term. original function We get back the original function (shown in blue on the right), but it is not simply multiplied by a collection of constants, as shown in lavendar on the left. The epression to the left also contains the variable,. Thus it is shown that this wavefunction is NOT an eigenfunction of the kinetic energy operator. T Op Ψ k( Ψ ) 8. Evaluate whether or not the function above is orthogonal to the function below: α g := e First I will normalize the second function, which I will call ψ. ψ := N o g ( N o g ) = N o := N o =.4 g ψ := N o g ψ Just checking that the function is normalized.

8 Eample_QM_calculations.mcd page 8 To check whether or not two wavefunctions are orthogonal I see if the orthogonality condition is satisfied: ψ a ψ b = 0 Here is the general orthogonality condition for two different wavefunctions, ψ a and ψ b. ψ ψ = 0 The two functions are orthogonal because the orthogonality integral equals What is the effect of changing α on each calculation above? Here is a sample student answer that shows substantial thought: I changed α (although I could not figure out a way to set up my template so that it would change the N value easily). I increased alpha to 0 then decreased to. I recorded on paper the following changes: α = 3 α = 0 α = N 0 = N 0 =.336 N 0 = 0.75 prob = 0.39 prob = prob = 0.6 over a small integral from = 0 to = 0.5 ep value = 0 ep value = 0 ep value = 0 σ = σ = 0.4 σ = I also noticed that the graphs changed. As you increase α the peak was more intense. As you decrease the value for α the peaks decreased in height. When the number e in the function is rasied to a smaller eponent (larger negative eponent) then the Normalization constant must increase, the probability of finding the particle in a small region near = 0 increases, the epectation value remains zero (the change did not shift the probabilty density away from it's symmetry about zero), and the standard deviation decreased (evident in the peak distribution being narrower, right?) When α was decreased the eact opposite occurred (epectation value was still zero though). I would have epected that since I increased the value for α more significantly than I decreased it that I would have seen more substantial changes for the increased value than for the decreased one. This was not really so. The standard deviation was greatly effected when I decreased α by only two units. The distribution of the graphs was much different (wider) than I epected.

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