SquareWell

# SquareWell - Main Page 1 Schrödinger’s Equation for the...

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Unformatted text preview: Main Page 1 Schrödinger’s Equation for the Finite Square Well The "look for bound state" button activates a macro, which adjusts the energy to minimize the value of the wave function at the extreme right, thus locating bound states. For a guide to using this spreadsheet, see Sheet 3. Energy of particle: E= 12.16726 4055.753 (Energy measured from bottom of well.) Adjust Energy with Slidebar! Depth of Well: D = 20 Width of Well: W= 4 Begin to integrate from x = Initial wavefunction value f(0): 5 Initial derivative value f'(0): Numerical step size dx: 0.01 Wavefunction at far right:-6426.2 the wavefunction changes from oscillatory to exponential. (The potential beyond the edge of the well may be too high to see on the graph.) This spreadsheet integrates f"(x) = (V(x)-E)f(x) with V(x) a finite-depth square well. We take V ( x ) = 0 inside the well, V ( x ) = D outside the well. NOTE 1: The potential V(x) - E , where E is the energy of the wavefunction, is plotted in green. At the point where this goes from negative to positive, NOTE 2:The wave function is plotted in red, the x-axis corresponding to ψ = 0. NOTE 3: If you click and hold on the end of the slidebar, the energy will change continuously. 1 2 3 4 5 6 7-10-5 5 10 Square Well Wavefunction Wavefunction Potential - E T a b le Pa ge 2 N U M E R IC A L SO L U T IO N O F SC H R O D IN G E R 'S E Q U A T IO N T he w a v e func tio n a nd its s e c o nd d e riv a tiv e a re fo und a t inte rv a ls n*d e lta _x , w he re n is a n inte g e r, a nd the firs t d e riv a tiv e is fo und a t the "le a p fro g " p o ints , (n + 0 .5 ) *d e lta _x . S o the num e ric a l s te p find ing the c ha ng e in the w a v e func tio n o v e r a n inte rv a l d e lta _x a s um e s it is d e lta _x m ultip lie d b y the v a lue o f the d e riv a tiv e a t the m id p o int o f the inte rv a l. T his is m uc h m o re a c ura te tha n us ing the v a lue o f the d e riv a tiv e a t the b e g in ing o f the inte rv a l. S im ila rly , to find the c ha ng e in the d e riv a tiv e o v e r a n inte rv a l the m e tho d us e d the v a lue o f the s e c o nd d e riv a tiv e in the m id le o f tha t inte rv a l. Position StepPotential Wavefunction 2nd_derivderiv at 1/2Potential - E v(x)-E 5 -60.8363-0.30418-12.16726 0.01 4.9 695819 -60.79 3-0.91217-12.16726 0.02 0 4.9878364 2 -60.68 3-1.51906-12.16726 0.03 4.97264587 -60.5035-2.12409-12.16726 0.04 0 4.95140495-60.245-2.72654-12.16726 0.05 0 4.92413953 -59.913-3.32568-12.16726 0.06 4.8908 27-59.5086-3.92076-12.16726 0.07 4.85167516 -59.0316-4.51 08-12.16726 0.08 0 4.80656439 -58.4827-5.0959-12.16726 0.09 0 4.75 60534-57.8627 -5.67453-12.16726 0.1 4.698 60 3-57.172-6.24625-12.16726 0.1 0 4.63639749 -56.412-6.81038-12.16726 0.12 0 4.56829373 -5 .5836-7.36 21-12.16726 0.13 4.4946316 -54.6873-7.91309-12.16726 0.14 0 4.415 0 74 -53.7245 -8.4503-12.16726 0.15 0 4.3 09 743 -52.6964-8.97 3-12.16726 0.16 0 4.2412 4 8 -51.6041 -9.493 4-12.16726 0.17 0 4.146291 2-50.4 9-9.9 783-12.16726-12....
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SquareWell - Main Page 1 Schrödinger’s Equation for the...

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