P 3 Qualitative Plots

8/13/2019 P 3 Qualitative Plots

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|| Om Shree Ganeshaya Namah |||| Om Shree Swami Samarth ||

PANEL-3

Question:Can we sketch the wave function out even before we attempt solve the SchrdingerEquation?

Can one sketch out the wave function without going through the grueling task of

solving the Schrdinger Equation? Yes! One can glean out a wealth of information

absolutely qualitatively and the tools to accomplish this are very elementary. What we

will be set up on carrying out is qualitative plots of wave functions! Let us re-iterate, it

is simple, but after we familiarize ourselves with some preliminaries.

Portrayed right below are two curvesin the y x plane, corresponding to two

different functions )( x y : let us call them )(1 x y and )(2 x y . Let us focus on the behavior

of the curves in the domain ],[ ba on the x-axis.

Tangent-2

y y1( x)

Tangent-1P Q

y2( x) Tangent-2

Tangent-1

1

2 x

a b

Fig. 3.1 The functions )(1 x y and )(2 x y and the tangents drawn to them at the designated points.

It requires no particular training in coordinate geometry it is manifestly evident to

recognize that the top function is more curved than the one below, within the x-domain

considered. Let us quantify this. Consider the top curve, )(1 x y . Draw a tangent at the

point P as depicted. Traverse the curve, i.e. take a path along the curve , and stop at the


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point Q, and draw a tangent at Q. Let denote, generically, the angle made by the

tangent to the curve with the positive direction of the x-axis. The curvature associated

with a curve at a point is the rate at which the tangents drawn would turn (i.e. the

change caused in the angle ) as one traverses along the curve. Let us denote the

elemental length 22 )()( dydxds ++= . We are considering, magnitude-wise, the

tangent turning rate i.e. the rate of change of the angle , with the curvilinear length s ,

i.e. | dsd / |. Just a look at the bottom curve ) (2 x y will clearly bring out that a similar

exercise carried out on it will lead to a smaller value for the rate of change | dsd / | .

The angle between the tangents 1 and 2, is greater than that between tangents 1 -2 .

Note that the tangents 1 and 1 have been drawn with the same value of the abscissa and

2 and 2 have been also drawn further ahead but again at the same value of the abscissa.

The following string of arguments can readily be invoked:

It is well-known that the slope of the tangent at a point is =dxdy

)tan( . Now,

ds

d

ds

d = )(sec)}{tan( 2 . (3.1)

But the left side of the above equation is

22

2

222

2

2

2

) / (1

1

)()( dxdydx yd

dydx

dxdx

yd dsdx

dx yd

dxdy

dsd

+=

+=

=

. ( 3.2)

Recognizing that )(sec 2 =2

2 1)(tan1 +=+

dxdy , equations (3.1) and (3.2) together

yield the formula for curvature that includes its algebraic sign now:

2 / 32

22

}) / (1{ / dxdydx yd

dsd

+= . (3.3)


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Note that the curvature has the dimensions of L-1 (distance) -1. The magnitude of the

reciprocal of curvature is the radius of curvature . The following is intuitively obvious:

The more curved a given path is the smaller is the radius of curvature and mutatis

mutandis (vice-versa). Concentric circles in a plane become less and less curved with

increasing radii; for the circle, the radius of curvature is constant and equals its own

radius. As an extreme case, a straight line segment has zero curvature and hence the

radius of curvature is infinite. When we negotiate a curve while driving a car, we must

supply the required centripetal acceleration / 2v , where v is the speed and the radius

of curvature. The sharper, i.e. more acute the curve is, the greater is the requiredcentripetal acceleration. If it is not supplied (say, through banking of roads and/or

through friction), the vehicle skids astray. A straight path with no curvature for nonzero

speeds has no centripetal acceleration meaning it has an infinite . Incidentally, a circle

of curvature for a planar smooth curve is constructed thus: Choose three distinct

neighboring non-collinear points on the curve. Let the extreme two points approach the

middle one so that the points are only infinitesimally separated. A unique circle drawn

through these points is the circle of curvature whose radius is precisely the radius ofcurvature! It changes from point to point in general.

Let us now consider what relevance this discussion can possibly have with the wave

function. Notice that the time-independent, one-dimensional Schrdinger equation that

incarnates as an Eigenvalue equation has a second-order spatial derivative. Let us re-

express time-independent, one-dimensional Schrdinger equation

)()()(2 2

22

x E x xV dx

d m

=

+

h (3.4)

in the form


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( ) E xV mdx

xd x

=

)(

2)()(

122

2

h

. (3.5)

We have been assuming real wave functions. For complex valued wave functions, we

must take the real and imaginary parts separately, since these two satisfy the same

Schrdinger equation, vide the linearity of the latter. In a compact notation, the left side

of Eq.(3.5) is )( / )( x x , which is well-defined except at the nodes of the wave

function )( x . The sketch depicted below has some curves drawn:

y

x

Fig. 3.2. Illustrations of some planar Convex functions.

What do the functions have in common? It will be immediately apparent that each

function, when viewed from the x-axis, is a convex function. Convex, meaning for the

functions in the upper half plane, any chord drawn always lies above the function except

at the end points (where it exactly meets the function and is therefore equal to it). For the

functions in the in the lower half, the interim points of a chord drawn lie below the curve.

In both the cases, the curve lies on the same side of the x-axis with respect to the chord.

Exactly opposite happens for a concave function, again of course, as viewed from the

x-axis. For concave functions in the upper half-plane, the interim points of the chord


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viewed from the x-axis always lie below the curve; and in the lower half, above the

curve.

y

x

Fig. 3.3. Some Concave functions in the plane.

Here, the segment of the curve lies on the opposite side of the x-axis with respect to the

chord. In particular, we are able to unequivocally relate the ratio of the second derivative

to the (real-valued) function for both convex and concave cases. To this end, we simply

perform the following construction for a typical segment of a convex function, such as a

local minimum, as shown in Fig. 3.4. Plot the function, its first and then its second

derivatives. The first derivative is negative initially, goes through zero, as it must, for an

extremum --here local minimum-- and then rises. The second derivative, which is the

derivative of the monotone increasing first derivative, is therefore positive in the chosen

domain ( cf . Fig. 3.4)


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)( x

x

)( x

x

)( x

x

Fig. 3.4. A segment of a function with a local minimum; hence (locally) convex. Note that the sign of thesecond derivative to the function itself is positive over the segment chosen.

Perform the same exercise for each convex function depicted, and you reach a moral that

the the ratio )( / )( x x is always positive for a convex function!


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Exactly the opposite is the case with the case of concave functions. Let us breeze

through the arguments analogous to the ones invoked above. For concave functions

residing in the upper half plane (hence non-negative), around a local maximum say ( cf.

Fig.3.3, top curve), the first derivative diminishes from initially being positive, going

through zero (as it must, precisely at the local maximum) and thence goes negative. Thus

the first derivative is monotone decreasing, which makes the second derivative negative

for the segment chosen. The ratio of the second derivative to the function is this time,

negative . In similar fashion, for the bottom curve in the lower half, the ratio second

derivative (positive) to the function (negative) is negative .

We thus have a thumb rule:

Function type (as viewedfrom the x-axis)

The ratio )( / )( x x

Convex Positive

Concave Negative

Transition fromconcave to convex

or convex to concave(Inflexion)

Zero(at the point of inflexion)

Table 3.1 Function (segment) types and the behavior of )( / )( x x

We have added a third row, for good reasons. It will answer the question: what if a

function makes a transition from convex to concave or concave to convex? At that point,the point of inflexion, the second derivative and hence the ratio is zero. A

curve that has the second derivative zero in some finite domain is a straight line:

B Ax x +== )(0 .

We map these inferences onto the quantum domain now. Referring back to Eq. (3.5),

it is clear that the ratio )( / )( x x is the that would discern on the convexity or

)( / )( x x


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concavity of the wave function )( x . In turn, it is the algebraic sign of the quantity

at different locations x that decides the nature of the wave function. We can

re-cast Table 3.1 as follows; cf. Table 3.2:

Sign of hence thatof the ratio Classically?

The Wave Function (asviewed from the x-axis)

Positive

Forbidden,Kinetic Energynegative, hencespeed: pureimaginary

Convex , either Monotoneincreasing OR decreasing;

OR Exactly one maximumor minimum; bulging

toward the x-axis

NegativeAllowed;Kinetic Energypositive, hencespeed: real

Concave, oscillatory withpositive and negative valuesin succession

Zero(at the point of inflexion)

Signifies aClassicalTurning Point:The particle

must turnback intoV ( x) < E region!

Transition fromconcave to convexor convex to concave(Inflexion) implying linear

behavior around theinflexion point

Table 3.2. Nature of the wave function with regard to the relative sign of .

Note that a smooth concave function cannot have more than only one type of extremum

in succession that is of the same algebraic sign, because, if it did have, then it would in

the middle bulge toward the x-axis which implies local convexity! To wit: two

successive local maxima A and B must be joined smoothly, and in the plane the onlymanner in which this can happen is that the curve ought to have a local minimum, C.

( ) E xV )(

( ) E xV )()( / )( x x

( ) E xV )(


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y BA

C

x

DFig. 3.5. Concavity implies oscillations: successive positive maxima; negative minima.

Now if we demand that the entire curvilinear segment ACB have the same algebraic sign,

i.e. is either exclusively positive (as depicted) or exclusively negative, then there will

inevitably be a convexity at C (viewed from the x-axis)! This is forbidden! Hence a real-

valued concave function must necessarily be oscillatory. Thus a proper, concave

function would be the curvilinear segment ADB; note that there is a change in sign--- the

function is positive at A and B while the intervening curve at the point D it is manifestly

negative. At all the locations A, D and B however the function is concave. Moral of the

story, then is: A real (1-D) concave function ought to be oscillatory with successively

occurring maxima and minima!

The foregoing discussion has established a crucial connection. The curvature is

connected with the second derivative, with the sign related to the ratio of the second

derivative to the function, cf. Eq. (3.3). The Schrdinger equation re-cast in the form of

Eq. (3.5) has precisely the ratio appearing on the left side and the quantity ( ) E xV m )(

22h

to the right. Since V T E += , The quantity ( ))( xV E is actually, classically, the


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local kinetic energy )( xT . For a classically permissible motion this quantity must be

non-negative (also called positive semi definite) i.e. positive, or in the least, zero ( it

exactly vanishes at a classical turning point). Negative values of the local kinetic energy

classically would mean the momentum (or velocity) is pure imaginary, making real

physical motion classically strictly forbidden. This analysis vindicates Table 3.2.

Enter Quantum Mechanics now!

Consider the de Broglie connection

ph / = (3.6)

When the function is oscillatory i.e. concave, the wavelength is real, meaningful. The

momentum )( x p p is (magnitude-wise) the local momentum. Since

( )V E m

p =2

2

V E m p += 2 ; (3.7)

which from the de Broglie relation ph / = gives,

)(2 V E m

h ph

== . (3.8)

The implication of Eq. (3.8) is now evident: The greater the difference E-V , the higher is

the momentum and smaller is the de Broglie wavelength. Again, recall that we are

considering only the classically permissible motion hence only concavity of the wave

function in a given region of space (the x-axis). This region could sometimes well be

considered a union of pieces, made up of different disjoint and/or overlapping

segments on the axis.

Consider as an illustration a simple potential (energy) function that is piecewise

constant and finite:


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V ( x)

E 1

E 2

xRegion I a Region II b Region III

E 3 )(1 x

I II III = I

x

Fig. 3.6. A simple potential energy distribution in one (Cartesian) dimension, with E 1, anEigenenergy in the continuum and E 2, a bound-state energy Eigenvalue. TheEnergy E 3 , below the minimum of V ( x) cannot be an Eigenenergy. The lowerplot is the qualitative sketch of the wave function

We notice that the for completely unbound state function ) (1 x with its characteristic

(Eigen-) energy E 1 has the quantity E V , i.e. here, ( ))(1 xV E is always positive and

further is piecewise (positive) constant. Hence V E is negative throughout and also

piecewise (negative) constant. Now this means that


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( ) 222

2

)(2)(

)(

1k const negative E xV

m

dx

xd

x

==

h

, say

Whence )()( 2

2

2

xk dx

xd

=

or 0)(22

2

=+ xk dxd

. Settingdxd

D ,

this gives [ ] 0)(22 =+ xk D ; for a nontrivial solution )( x that is not identically zero

everywhere, the operator preceding the function must vanish, yielding for D the solution

ik D = , which yields the two linearly independent solutions superposed to engender in

general

)cos()sin()( kx Bkx A x += , (3.9)

or, equivalently,

ikxikx

DeCe x

+=)( . (3.10)

This vindicates, for the present case in particular, that the concavity leads to sinusoidal

and hence oscillatory functions! Note however that we have divided the x-axis into three

regions I, II and III, so we would have the three wave vectors III II I k k k ,, and the

corresponding wavelengths III II I ,, respectively. Clearly, in the regions I and III, the

positive ( E V ) difference being exactly the same, the de Broglie wavelengths I and

III are equal, while in the interim region II, the corresponding wavelength

II is smaller

because of higher ( E V ) value there.

The aforesaid affirms that the wave function is oscillatory with a reduced wavelength

in Region II Reduced wavelength means that the peaks would crowd together relative to

those for the outside oscillatory function. But how could we now sketch the wave

function? Simple! Just draw a smooth portrayal, i.e. a wave function which is

continuous with its spatial first derivative also continuous (this last one is required since


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the potential is finite and has at best a finite number of finite discontinuities) throughout.

Note that we have two free, floating, arbitrary constants in the solution for )( x . We

can exploit the freedom and match the value and the slope at every point for a finite )( xV .

The pivotal points are the ones at the interface where there is a change in the form of .

V ( x)

E 2


Fig. 3.7. A qualitative sketch of the wave function corresponding to the a bound-state energy Eigenvalue E 2.

Let us now try to plot the wave functions for a bound-state and hence discrete energy

2 E . Note now that there is a qualitative difference in the nature of the wave functions in

the two extreme regions I and III, from the one that the middle region II flanked by I and


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III. Why is this so? Look at the sign of E V ! It changes from being positive (with a

constant value in the example above) in the extreme two regions I and III; to being

negative (with another constant value), in the middle region, II. In this case, the pertinent

state of affairs is summarized as follows. For Regions I and III,

( ) 222

2

)(2)(

)(1

==

const positive E xV

mdx

xd x h

, say

Whence )()( 2

2

2

x

dx

xd

=

or 0)(22

2

= x

dx

d . Putting

dx

d D ,

We now have 0)(22 = x D ; again, for a nontrivial solution )( x that is not

identically zero everywhere, the operator preceding the function must vanish, yielding for

D the solution = D , leading to two linearly independent solutions superposed,

expressible as, in general

x x DeCe x +=)( ; (3.11)

or alternatively, as

)cosh()sinh()( x B x A x += . (3.12)

Note now that the above indicates that the wave function in Regions I and III is a

superimposition of convex functions. The exponential functions with real arguments or

equivalently, the hyperbolic functions do not oscillate. The middle Region II has as

before a positive V E implying concavity, i.e. oscillatory behavior.

Now to depiction of the wave function. We have oscillations midway and exponential

behavior in the extreme regions. We appropriately choose a decaying function


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diminishing as x becomes more and more negative , i.e. on the left hence ~ xe for

negative x. We also demand that the wave function also diminish for increasingly

positive x which converges on the choice ~ xe for positive x. Why diminishing

functions for magnitude-wise large values of x? The alternative choice of increasing

functions would violate the finiteness of the wave function that is demanded from

normalization of a bound-state function. The constants can be adjusted to match the

value and the slopes at the interfacial points a x = and b x = .

Additional features that must be noted are the following: the rate of decay of the wave

function is with the same decay constant . Thus the rate of damping of the wave

function is the same on the two sides. This would be different if the potential had its

barriers with different heights in the Regions I and III.

How about the amplitudes now? Come on, they cannot be plotted pin-pointedly in a

qualitative sketch, for if we were able to, there would have been no need to solve the

Schrdinger equation (!)but it is not a totally hopeless situation--- Given a very high

excited state of energy (Eigenenergy), the Bohr correspondence principle plays its hand

and one could say that the averaged absolute square quantum mechanical wave function

mimics the classical probability distribution. Also, for a bound state, for a potential

energy function with no sharp (vertical) jumps, at the classical turning points, the

classical momentum is momentarily zero (and it gets reflected back to the classically

permissible domain), hence the particle would spend most of its time thereabout. Since

the wave function is the probability amplitude, it reflects this, but only for a high excited

state. For a very high excited state there would be more oscillations (the de Brogliewavelength will be small, successive peaks get closer), and a mean curve would be a

constant, coinciding with the classically uniform probability density.


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V ( x)

E


)( x

I a II b III

Fig. 3.8. A qualitative sketch of the wave function corresponding to a bound-state energy Eigenvalue E ,with a tilted linear potential midway with different but constant barrier heights on its sides.

Consider now the potential drawn in Fig. 3.8. We have here a linear potential sloping up,

which has two potential barriers with different constant heights on its sides. Note that

for this situation, in Region II between the classical turning points a and b, the difference

))(( xV E is positive, thus the wave function in that domain is concave, oscillatory. The

difference continuously decreases meaning the (local) de Broglie wavelength )( x

continually increases (in the sequence blue green red ) until it hits the right classical

turning point. Subsequent to that we must have a monotone decrement (by virtue of its

convexity). The same goes for the left hand classical turning point: there will be


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we compromise finiteness, the value and slope matching goes haywire! Moral: if we

demand both these to match, then the interim region must have concavity , i.e. there must

be at least some finite segment of the x-axis with E > V ; which is a classically

accessible domain. For an acceptable wave function, V cannot exceed an allowed

Eigenenergy E everywhere ! See the figure below to have this manifest:

)( x

x

Fig. 3.10. A smooth, permissible wave function.

Actually, a similar line of reasoning can be procured to understand why bound states

are associated with discrete energy levels. The wave function corresponding to the

lowest energy Eigenvalue, i.e. the ground state energy Eigenfunction has no interim ---

i.e. discounting those at --- nodes. For a slightly higher energy, a greater support for

the region(s) E > V is offered meaning that there will be more oscillations.

Fig. 3.11 has the potential energy plotted that supports some bound states. We start

off with the ground-state function corresponding to the energy E 1. We try to plot a

function for slightly higher but disallowed energy value = E . How? Look at the

asymptotic nature of the wave function, that will be convex and should increase as one

approaches the active region of the well from both the extremities ( and ) . The

moment one encounters the region E >V , greater degree of concavity sets in, and

)(1 x


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oscillations occur. However, this leads to a mismatch: either in slopes while the values

match as in the top continuous curve on the right; OR matching slopes but not values (the

bottom dashed curve, obtained by scaling and reflecting the right hand side solution) , cf.

the wave (but not Eigen--) function )( x .

We sweep the energy to higher values, where it should be evident that still greater

degree of concavity could yield a profile as for )(2 x . In the process, the wave

function passes on the other side of the x-axis generating a node. You can continue the

process and observe that the continuity of the function and its slopes leads to only a

certain set of allowed (Eigen-) values! Please refer to Schiff for further reading.

V ( x)


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x

E 2

E E 1

values and slopes match

CTP CTP

values match, but not slopes

)( x

slopes match, but not values)(2 x

values and slopes match; additional node generated

Fig. 3.11. Origin of discrete energy Eigenvalues for bound states: plausibilityarguments. The sideways vertical lines designate the classical turningpoints (CTPs).

)(1 x


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The number of nodes increases as the energy Eigenvalue increases. Once again, this

is a direct consequence of higher concavity hence ups-and-downs. However, the kinetic

energy expectation value which is, for a bound state proportional to the absolute square

of the derivative of the wave function is also expected to enhance. This requires a higher

rate of change of the wave function, again consistently, higher oscillations hence more

nodes!

Please refer to Crasemann and Powell for the Sturm-theorem on the nodes of the

Eigenfunctions: higher energy means more nodes. In fact, there is a stringent interlacing

of nodes theorem that between two successive nodes of a given Eigenfunction, there isexactly one node of the immediately succeeding excited energy Eigenfunction!

We now present below some potential (energy) distributions in 1D, and the

corresponding energy eigenfunctions plotted qualitatively.


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V ( x)

E

x

E-V E-V +ve, diminishing E-V E-V negative -ve const +veconst const

)( x

x dampedconvex

concave,oscillatory, increasing convex, osc

diminishingfor incidence from left

Fig. 3.12. A qualitative portrayal of the wave function. Note the nature in differentRegions with regard to convexity and concavity.


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V ( x)

E

x

)( x , symmetric

x

convex

damped oscillatory oscillatory dampedconvex increasing decreasing convex

)( x , anti-symmetric

Fig. 3.13. For a symmetric (even) potential: Non-degenerate (1D bound states are non-degenerate) have adefinite parity : they must be exclusively symmetric (even, top curve) OR antisymmetric (odd, bottomcurve) in x.


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Let us next consider a bound state for a symmetric potential i.e. when V ( x) is even

function of x. There is a theorem that in one dimension, bound states are always non-

degenerate ( cf ., e.g. Crasemann and Powell). For this potential, it is strictly required that

a non-degenerate bound state ought to have a definite Parity : they must be either an even

function of x, or, exclusively, an odd function of x. For this potential, Fig. 3.12

represents these two possibilities, in addition to the standard constraints. Note that a

symmetric convex function must exhibit a local minimum if positive or a maximum if

negative. For bound stats, everywhere in the extreme regions we have labeled the

function: damped function, which is in the spirit of the absolute square being a

diminishing function.

The interested reader might keep wondering: when exactly should we divide the axis

into regions and how? Simple, again!

Whenever

(i) Whenever the quantity ( E V ) changes its algebraic sign

(ii) Whenever the potential V ( x) changes its functional form and

(iii) Whenever the potential suffers a discontinuity.

(iv) For infinite discontinuity in V ( x), the derivative of the wave function must

necessarily suffer a discontinuity.

Observe that we have, in the figures, subdivided into different segments exactly in

consonance with these maxims!

It indeed is gratifying that it is possible to pre-emptively stipulate the form of the wave

function qualitatively with regards to its convex or concave nature. Quantification is

possible only for a very high excited state, for which the average quantal probability

density mimics the classical one, as an offshoot of the Bohr correspondence principle.

A comment for positively infinite potential would be in order. At the infinite

discontinuity, the wave function ought to suffer a derivative discontinuity. This follows

from integration of the Schrdinger equation in a small region around the infinite

discontinuity. If the potential (potential energy function) is positively infinite in a region


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(finite or semi-infinite line segment say), then the wave function gets guillotined at the

interface and ought to vanish in the region of the segment.

There exists an attractive delta-function potential that is negative singular at one point

only and vanishing elsewhere. There, the wave function will be continuous while the

derivative will perforce be discontinuous, entailing a cusp. For the solution to this

problem, see Griffiths. Professor Dr. T. Padmanabhan (IUCAA, Pune) has devised a

nifty way to transform the attractive -function well problem to the momentum space and

solve it, and take the Fourier transform yielding the co-ordinate space wave function.

The procedure is straightforward (once you get to know it wisdom of hindsight!) onceit is recognized that the -function has a constant Fourier transform in the counterpart-

space in conjunction with the convolution (Faltung) theorem.

If in a region (in 1-D) the binding potential is piecewise constant and the quantity ( E

V ( x)) negative, then the convex wave function is exponential, appropriately with a

diminishing modulus. However, a linearly increasing or super-linearly increasing

potential (respectively for a uniform force field (linear potential) like in the case of a

harmonic oscillator (with a quadratic parabolic potential), the decrement of the

wave function is faster, superlinear in the exponent . The decaying part is Airy function-

like: )( x ~ )exp( 2 / 3 x (cf. Landau and Lifshitz, Crasemann-Powell, Griffiths) for a

linear potential (with nonzero slope) and has a Gaussian damping for the oscillator

potential )( x ~ )exp( 2 x (apart from a ( Hermite --) polynomial factor).

Further Reading : Crasemann-Powell, Schiff

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