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COMPlEMENT H"
Complement Hx,
EXERCISES
T/1¡(EA ONMLKJIHGFEDCBA1 1 baZYXWVUTSRQPONMLKJIHGFEDCBA'1 ZYXWVUTSRQPONMLKJIHGFEDCBA
." " '" t.y! A particle of mass m is placed in an infinite one-dimensional well of width a :
V ( x ) = O
V ( x ) = + 00
for O!( x !( a everywhere el se It is subject to a perturbation W of the form:
where 1110 is a real constant with the dimensions of an energy. a. Calculate, to first arder in \v a ' the modifieations induced by W ( x ) in the energy levels of the particle. b. Actually, the problem is exactly soluble. Setting k = - J 2 m E jñ 2 , show that the possible values of the energy are given by one of the two cquations sin ( k a /2 ) = O or tan ( k a /2 ) = - ñ 2 k /m m A ,! 0 (as in exercise 2 of complerncnt LI, watch out for the discontinuity of the derivative of the wave function at x = a i2 ). Discuss the results obtained with respect to the sign and size of \V (^) o ' In the limit \v o - - - - - O, show that one obtains the results of the preceding question.
- Consider a particle of mass In placed in an infinite two-dimensional potential well of width a ( e l complement G Il) :
V ( x , y ) = O
V ( x. y ) = + 00
ir o!( x !( { 1 and O!( Y !( a everywhere el se This particle is also subject to a perturbation W described by the potential:
W ( x , y ) = 11'0 for^ O!(^ x^ !(^ t¿^ and^ O!(^ Y^ !(^ a 2 2
W ( x , y ) = O everywhere else
a. Calculate, to first arder in \v a ' the perturbed energy of the ground state. b. Same question for the first excited state. Give the corresponding wave funetions to zeroeth order in w (^) o.
1 2 0 0
E X E R C IS E S -------------------------------------------------------------------- baZYXWVUTSRQPONML b. wvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBABy finding upper bounds for the terms of the series for ONMLKJIHGFEDCBAf: 2 , give an upper bound Ior E 2 ( e l §B-2-c of chapter XI). Similarly, give a lower bound 1: 2 , obtained by retaining only the principal term of the series. With what accuracy do the two preceding bounds enable us to bracket the exact value of the shift L lE in the ground state to second order in g?
c. \Ve now want to calculate the shift L lE by using the variational method. Choose as a trial functíon:
where « is the variational para meter. Explain this choice of trial íunctions. Calculate the mean energy < H ) ( IX ) of the ground sta te to second arder in g [assuming the expansion of < H ) ( a ) to second order in 6 to be sufficient]. Determine the optimal value of a. Find the result ¿ J E var given by the variational method for the shift in the ground state lo second arder in rff. By cornparing ¿ J E v a r with the results of b , evaluatc the accuracy of the variational method applied to this exarnple. We give the integrals:
-^2 r^ a^ (^ x - - - - a ).^ SIll ( n x ) -^ SlIl_._^ ( 2 n n x ) ---- dx = a.o 2 a a n = 1,2,3, ...
" ) f" ( )2 () '( ) _~ a __ 2 n x a - 1 1
- x - - :; Si11 - - dx = - - - - - a o ~ ( l 2 6 t:
a.^2 r^ o a^ (^ x - " 2a). Sl11 ( n x ) -;,-;.^ COS ( 7 [ x ) ---;^ dx ,^ = - 2 na
1 6 n a
For all the numerical calculations, take n 2 = 9.87.
~ '\
9.) We want to calculate the ground state energy of the hydrogen ato m by the
- ,hiational method, choosing as trial functions the sphericaJly symmetrical functions (P.(r) whose r-dependencc is given by:
for r ~ o :
for r > 'l.
e is a norrnalization constant and 'l. is the variational parameter.
a. Ca1culate the mean value of the kinetic and potential energies of the electron in the state I q J. ). Express the mean vaJue of the kinetic energy in terms
Timé llldepéllJtllt Pcrturbarion Theory 267 ONMLKJI
V ( .Y ,baZYXWVUTSRQPONMLKJIHGFEDCBAy , z )
----.-- .. ------->--z --- (^) --- .,,------
- /,.,.... ZYXWVUTSRQPONMLKJIHG/ - - _ _ - t~ ~ ~ Z I^ /^ - I^ I / I /^ / c^2 (.~·2 + y : ) - - ¡ - 'z .2 ~ : ,
Fig. 16-2, Schematic picrure of poremial entrgy as a funcrion of z wirh x and y bcld fixed. The dorred line represenrs rhe Coulornb porential, rhe dashed line thc porenrial energy due to rhe externa! tield, and tbe salid line the total potential.
however, rhey may be stable on a time scale of the age of the universe," and hence the observarions agrce perfecdy wirh what the firsr few terms of the pcrrurbarion series predice
Problems .J'
\ 1.) Consider rhe hydrogcn arorn, and assume thar the protan, insread of being"a: point-source of the Coulomb field, is a uniformly charged sphere of radius R , so rhat rhe Coulomb potencial is now modified to
. 3 e 2 V e r ) = - o - (^ R 2 - ... 1) r 2 2 R ".
r < R ( < < a o )
e -^ ., ,.^ ,. >^ R
Calcularé (he energy shift for the n = 1, 1 O sta te, and for (he n = 2 stares, caused by chis modificarion, using rhe wave funcrions given in (12-25).
, Acrually a simple barrier penetratio a calculation of th e rype carried o u t in Chaprer :; shows rhar the rime scale is more like 10 1 0 0 1 ) liferirnes of rhe universe, for fairly reasonable fields!
268 Quanrum PhysiesZYXWVUTSRQPONMLKJIH ,~
(2.: Calculate the energy shifr in the ground stare of (he one-dimensional harmi5riic oseilJator, when the perturbation baZYXWVUTSRQPONMLKJIHGFEDCBA V = A X 4 15 added (Q
- Consider a square well in one dimcnsion. If rhe edges ONMLKJIHGFEDCBa f thc well are rounded off as shown in the figure, what is the change in the graund state energy? r m Choose your rounding-off pararnctrization such rhat J -r co V ( x ) d x rernarns un- changed.
1 J
..~
(4. .The borrorn of au infinite well is changed to have the shape J 1 fX V ( x ) = E sin - (^) b O < - - x < b
Calculate the energy shifts for all the excited states to first arder in E. Note that rhe well originalJy had V ( x ) = O for O ~ x ~ b , with V = 00 elsewhere.
- Prove (he surn ruJe (Thomas-Reiche-Kuhn sum rule) f¡? L ( E n - E a ) I ( n lx la ) i2 = - n 2 m
[ H in t. (a) Write the cornmutation relation [ p ,x ] = ñ ji in the form
L f ( a lp ll2 ) ( n lx la ) - ( a lx ln ) ( n IP la ) l = ~ ( a la ) = h n 1 J t (b) Use the faet that I n i - ( a llH ,x ] ln ) fí
in wocking out rhc problern.]
- Check the above sum ruJe for rhe one-dimensional harmonie oscillaror, wirh " a " taken in rhe ground stare.
- Work Out the firsr arder Srark effeer in rhe 12 = 3 srare of the hydrogen atom. Do not borher to work out all the integrals.
- Consider an electrón in a state 1 1 in a hydrogen atorn. The atorn is placed
A P P R O X IM A T IO N M E T H O D S F O R S O U N D S T A T E S wvutsrqponmlkjihgfedcbaZYXWVUTSRQP 295
generating function (13.10) for the Hermite polynomials. The earlier discussion (Sec. 13) shows that the states most likely to be excited are those that have a classical amplitud e of oscillation that is of the order of the displacemcnt ONMLKJIHGFEDCBAa ; this is in agreement with the correspondíng classical rcsult. Equation (35.32) can then be used to show that the sud den approximation is valid in this case if the time rcquired tu move the equilibrium point is small in comparison with l/no times the classical- oscillator period, where baZYXWVUTSRQPONMLKJIHGFEDCBAn o is the quantum number of the state most likely to be excited.
PROBLEMS l. A one-dimensional harmonic oscillator 01 charge e is perturbed by an electric field 01 strength E in the positive x direction. Calculate the change in each energy level to second order in the perturbation, and calculate the induced clectric di poi e momento Show that this problem can be solved exactly, and compare the result with the per- turbation approxirnation , Repeat the calculation for a three-dimensional isotropic oscillator. Show that, ir the poJarizability exof the oscillator is defined as the ratio of the induced electric dipole moment to E , the change in energy is exactly - t" ,E '.
- A one-dimensional harmonic oscillator is perturbed by an extra potential energy b x ". Calculate the change in each energy level to second order in th e perturbation.
- Find tho first-order Stark effect for a hydrogcn atom in the stat.e n = 3. Sketch l. j.~e _- _^ arrangcment^ of the^ lcvcls^ and^ st.ate^ the^ quantum^ numbers^ associatcd^ with^ each.
~
·.4../¡ A system that has th ree unperturbcd states can be representad by the pcrturbed
. amiltonian matrix
where E , > E ,. The quantities a and b are to be regarded as perturbations t.hat are 01 (he same order and are small compared wit.h E , - E ,. Use th e second-order non- degenerate perturbation theory to calculate the perturbed eigenvalues (is this pro- ccdure correct?). Then diagonalize the matrix to find the exact eigenvalues. Finally, use the second-order degenerate perturbation theory. Compare the three results obt.ained.
- A trial function f differs from an eigenfunction U E by a srnall amount, so that f = U E + .'h , wher e U E and f, are normalized and • «1. Show that ( H ) differs from E only by a term of order ." and find this termo
- Ir the first n - 1 eigenfunctions 01 a particular hamiltonian are known, write a formal expression for a variation-rnethod trial function that could be used to get an upper limit on thc nth energy level.
- Find the next terms (01 order R - ') in the cxpansion of Eq. (::12.12). Show that thcir diagonal matrix elcment for the unperturbed ground statc vanishes, so that thcrc is no inverse fourth-power contribution to the van der Waala interaction.
- Use the first nonvanishing term in the series (32.13) to get a lower limit for - W ( R ). Compare with that obtained from th e variation ealculation.
