How can we confine an electron?
name of author
Electrons are particles of negative charge.
To keep them in one place, rather than wandering everywhere, you can use attractive or repulsive forces.
Early ideas- the Bohr atom- thought of them as classical particles orbiting a central nucleus, and explained the existence of discrete energy levels, rather than a continuum, by assuming an associated wave had to fit round each orbital circumference.
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Building blocks- electron, potential well
Electron- fundamental particle, charge 1.6E-19 coulombs. Mass
Electric potential- work neeeded to get to the point being described from some origin, often from infinity or a local reference point.
Measured in volts, which are joules-per-coulomb.
Potential wells are like rubber sheets. Stretch a rubber sheet on a horizontal frame, and pull it down in the middle. The motion on this curved depression of a ball bearing pulled by gravity are SIMILAR ( but not identical!) to electrons moving in a potential well of electrostaic fields.
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Electron as a wave/particle
The electron is described in modern mathematical physics by a wave equation using complex ( 'imaginary' ) numbers.
Its motion has to be described by the mathematics of 'complex numbers.
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Confining a wave
Simplified explanations mix particle and wave metaphors.
A GRAVITY potential well allows any energy to be the current state of the orbiting particle. The planets could orbit in any of a continuous series of radii, and also in varied elli[ses. No wave behaviour is seen on this scale, 'tho we have to allow for small relativistic effects.
For an electron we can only expect certain energies to be possible, so that the 'wave fits'. We can't avoid the wave aspect.
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Form of a single wave. ( and end-point restrictions)
For a simple confined small charge a differential equation describes the form of behaviour.
The resulting curves, in complex-number form, are interpreted in terms of probabilities of where the electron will be found at any given time.
It has to be taken along with end-point restrictions. The most obvious is that there has to be zero probability of finding it beyond a certain radius and the other is that at the centre we have either zero probability, giving anti-symmetric answers, or finite probability but zero gradient, giiving symmetric results.
The differential equation spells out how the graph amplitude varies with position, by stating how the acceleration depends on radial position, radial acceleration, and energy at that position in the potential well.
Such equations were created by Erwin Schrodinger 90 years ago.
Technically speaking we are looking for 'eigen solutions.'
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Exploring and discounting possible solutions
To home in on acceptable solutions we could keep trying different energies and get radial distribution values for each as a column of numbers.
Can you 'see' if we are near a solution distribution?
We'd reject any that do not fit our end-point requirement ( tending to zero at high radii), and try to home in on those which do fit. So we'd reject this value of E and try another. Try E =0.0010. Is it better or worse? Tabular results are not easy to interpret!
This is all very SLOW for humans, even if the computer does the numerical processing! Feel very sorry for the mathematical physicists who had to produce such test solution tables by hand!
' schrodH2c.bas -
dim lastVals( 1000)
global x0, k, E
dx =0.05: x0 =10: k =0.35E-3: L =1
lastSense$ ="up"
E =0.001 ' <<<<<<< try changing this to get a better solution attempt!!
y =20: yDot =0: yDotDot =0
while ( x <50) and ( abs( yDot) <10^6) and ( abs( yDotDot) <10^6) and ( abs( y) <55)
x =x +dx
yDotDot =y *( V( x) -E)
yDot =yDot +yDotDot *dx
y =y +yDot *dx
print using( "##.###", x), using( "##.####", y)
wend
wait
function V( x)
if x >x0 then V =k *( x -x0)^2
end function
[quit]
close #w
end
GUI and automatically determining valid solutions
Fortunately we can use a GUI graphic interface and the power of digital stepwise solution to do it faster.
Instead of a column of numbers we look at each curve for specific energy states and can immediately see if it is near to being a solution.
Coding this allows us to see a range of energies and the resulting probability curves, and see if any were near a solution.
Watch for the curve 'flicking' between a + and a - infinity value at higer radii. A solution must exist in between these...
Try decreasing the step to get closer attemts at an acceptable solution.
' schrodH2c.bas -
nomainwin
WindowWidth = 530: WindowHeight =350
UpperLeftX = 20: UpperLeftY =120
graphicbox #w.g2, 10, 10, 500, 300
global x0, k, E
dx =0.02: x0 =10: k =0.35E-3: L =1
lastSense$ ="up"
Emax =0.4500
open "Hydrogen atom demonstrator 'schrodH2c.bas' johnf June 2016 (2003)" for window as #w
#w "trapclose [quit]"
#w.g2 "cls ; down ; fill 150 150 255 ; line 0 130 500 130 ; flush"
for E =0.00001 to Emax step 0.002
y =20: yDot =0: yDotDot =0
while ( x <50) and ( abs( yDot) <10^6) and ( abs( yDotDot) <10^6) and ( abs( y) <55)
x =x +dx
yDotDot =y *( V( x) -E)
yDot =yDot +yDotDot *dx
y =y +yDot *dx
#w.g2 "set "; 10 *x; " "; int( 130 -3 *y)
scan
wend
x =0
next E
wait
function V( x)
if x >x0 then V =k *( x -x0)^2
end function
[quit]
close #w
end
Comprehensive GUI
The best thing is to use the computer to detect for itself when it is very near an acceptable solution. NB it will never be exactly on a solution, since there are finite limits on how accurately we are computing.
Here I work through a series of possible energy states. Most 'go to infinity' in a up or down direction, so are not valid solutions.
The electron is unbounded and escapes.
The program looks for the 'whip lash' as we swap from a solution going to +infinity to one going to -infinity.
A bounded solution will exist between these energies, dropping to zero 'outside', and we could find a closer energy by fine tuning the energy. The 'close to solution' is replotted on the right, without the clutter of failed solutions.
These solutions describe a wave contained as a wave inside a finite-radius atom.
A series of solutions exists, at increasing energy states and with more complex radial electron distribution. See the right-hand pane.
These are drawn for each state at the right hand side.
' schrodH2c.bas -
nomainwin
WindowWidth =1240: WindowHeight =550
UpperLeftX = 20: UpperLeftY =120
graphicbox #w.g2, 40, 10, 500, 300
graphicbox #w.g3, 560, 10, 500, 500
graphicbox #w.g4, 1080, 10, 140, 500
statictext #w.st1 "", 40, 320, 500, 500
dim lastVals( 1000)
global x0, k, E
dx =0.05: x0 =10: k =0.35E-3: L =1
lastSense$ ="up"
Emax =0.4500
open "Hydrogen atom demonstrator 'schrodH2c.bas' johnf June 2016 (2003)" for window as #w
#w "trapclose [quit]"
#w.g3 "down ; color black ; size 1 ; fill 180 180 40"
#w.st1 "!font Arial 12"
#w.st1 "Attempts at solutions appear above. Electron probability vs radius." +chr$( 13) +chr$( 13) +_
"Most energy level/states tried produce the impossible, unbounded state" +chr$( 13) +chr$( 13) +_
"However, certain energies give a bound, stationary state for which" +chr$( 13) +_
" the probability is zero of finding the electron 'outside'." +chr$( 13) +chr$( 13) +_
"NB while we could home in on the exact values, I just look for" +chr$( 13) +_
" the curve flicking from heading violently to +/- infinity." +chr$( 13) +chr$( 13) +_
"Detected stationary states are copied on the right. ============ >>>>"
#w.g2 "cls ; down ; fill 150 150 255 ; line 0 130 500 130 ; flush"
#w.g4 "down ; fill 150 150 225"
for E =0.00001 to Emax step 0.002
'lastSense$ ="up"
count =0
y =20: yDot =0: yDotDot =0
R =int( 256 *E /Emax): G =0: B =255 -R
col$ =str$( R) +" " +str$( G) +" " +str$( B)
#w.g2 "color "; col$
while ( x <50) and ( abs( yDot) <10^6) and ( abs( yDotDot) <10^6) and ( abs( y) <55)
x =x +dx
yDotDot =y *( V( x) -E)
yDot =yDot +yDotDot *dx
y =y +yDot *dx
#w.g2 "set "; 10 *x; " "; int( 130 -3 *y)
lastVals( count) =y: count =count +1' save for redraw with XOR to erase??
scan
wend
flag =0 ' if flag =0 do not draw the point
if ( (y >0) and ( lastSense$ ="down")) or ( (y <0) and ( lastSense$ ="up")) then' (solution limit changed sign)
if lastSense$ ="down" then lastSense$ ="up" else lastSense$ ="down"
for j =count -1 to 1 step -1 ' plot previous saved values
if ( abs( lastVals( j)) <0.1) then flag =1' don't plot outer areas where always goes to infinity due to errors
if flag =1 then
col$ =word$( "255 0 0, 0 255 0, 0 0 255, 255 255 0, 0 255 255, 255 0 255, 0 0 0, 200 200 200", L, ",")
#w.g3 "color "; col$
#w.g3 "line "; int( 500 *j /1000); " "; 62 *L -30 -1.5 *lastVals( j); " "; int( 500 *j /1000); " "; 62 *L -30
last =lastVals( j) *4
if last >=0 then R =int( 128 +min( 127, last)) else R =int( 128 -max( -127, last))
c$ =str$( R) +" 0 0"
#w.g4 "color "; c$
#w.g4 "backcolor "; c$
#w.g4 "up ; goto 60 "; -40 +L *62; " ; down ; circlefilled "; int( j /30)
end if
next j
L =L +1
end if
x =0
next E
wait
function V( x)
if x >x0 then V =k *( x -x0)^2
end function
[quit]
close #w
end
CAVEAT
Please note that all I am doing here is giving a flavour of why electrons exist in an atom only at certain energy levels.
It is a presentation designed to be intelligible to a student at UK 'Sixthb Form' Level, ie 16+.