Quantum Chemistry : Particle in Box...Quantum Chemistry : Particle in Box FMIPA UGM – 02 March 2020 Niko Prasetyo 2 Application of quantum mechanics Translational motion –Molecules

Quantum Chemistry :Particle in Box

FMIPA UGM – 02 March 2020

Niko Prasetyo

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Application of quantum mechanicsTranslational motionTranslational motion– Molecules store energy as translation, rotationMolecules store energy as translation, rotation

and vibrationand vibration– In this section we will discuss the application of In this section we will discuss the application of

translational motion in box 1D and 2Dtranslational motion in box 1D and 2D

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Application of quantum mechanicsTranslational motionTranslational motion– Based on the postulat of quantum mechanics, the acceptable wavefunction is Based on the postulat of quantum mechanics, the acceptable wavefunction is

for region II.for region II.

• At region I and III, the potential energy is infinite, wavefunction is 0At region I and III, the potential energy is infinite, wavefunction is 0

– Thus, a boundary condition is applied to force the particle to move in region IIThus, a boundary condition is applied to force the particle to move in region II

• In this region, particle moves freelyIn this region, particle moves freely

• No influence of external forcesNo influence of external forces

• No go to outsideNo go to outside

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Application of quantum mechanicsTranslational motionTranslational motion– Schrödinger equation for particle in 1 DSchrödinger equation for particle in 1 D

– In region I and IIIIn region I and III

– In region IIIn region II

∂2ψ∂ x2

+ 8 π2 m

h2(E−V )ψ=0

∂2ψ∂ x2

+ 8 π2 m

h2(E−~)ψ=0

∂2ψ∂ x2

+ 8 π2 m

h2(E−0)Ψ=0 ∂2ψ

∂ x2+ 8 π

2 mh2

EΨ=0

Valid for

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Application of quantum mechanicsTranslational motionTranslational motion

If If = A sin ax, then = A sin ax, then

∂2ψ∂ x2

+ 8 π2 m

h2EΨ=0

∂2ψ∂ x2

=−8 π2m

h2Eψ

(− h2

8π 2m) ∂

2ψ∂ x2

=Eψ

(− h2

8π 2m)∂2 (ASinax )

∂ x2=EASinax

(− h2

8π 2m) .−Aa2 Sinax=EASinax

h2a2

8π 2mψ=Eψ

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Application of quantum mechanicsTranslational motionTranslational motion

In order to get value of a, we have to apply the interpretation of In order to get value of a, we have to apply the interpretation of

• has to be continue, single valued and finitehas to be continue, single valued and finite

•In region II, In region II, remember, not the probability density)remember, not the probability density)

∂2( Sinax)∂ x 2

=−a2 Sinax

E= a2 h2

8 π2m

= A Sin ax = 0

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Application of quantum mechanicsTranslational motionTranslational motion

Sin ax = 0, if ax = nSin ax = 0, if ax = n

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Application of quantum mechanicsTranslational motionTranslational motion

– The border of box, x = 0 and x = L, thusThe border of box, x = 0 and x = L, thus

a= nπLn = 1,2,3,……

En=n2h2

8mL2

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Application of quantum mechanicsTranslational motionTranslational motion– In region II, In region II, has to be normalized has to be normalized

∫0

L

ψ∗ψdx=1

∫0

L

ψ2 dx=1

∫0

L

(A Sinax )2=1

A2∫0

L

Sin2 ax dx=1

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A2∫0

L

Sin2 ax dx=1

x2−1

2 [ 12a sin2 ax ]a= nπL

∫0

L

sin 2axdx x2−1

2 [ 12a sin2 ax ]L2− 1

2 [ 12a sin2 nπL L]

=

=

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X = 0

x2−1

2 [ 12a sin2 nπL x ]02−1

2 [ 12a sin2 nπL 0 ]

X = L

x2−1

2 [ 12a sin2 nπL x ]L2−1

2 [ 12a sin2 nπL L] L2=

∫0

L

sin2

axdx= L2

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∫0

L

sin2

axdxA2 = 1

A2 L/2 = 1

A= ( 2L )12

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A=( 2L )12 ψn=( 2L )

12 Sin nπx

L

Probabilty to find particle is depends on x Probability to find is also depends on the energy, n Only certain values of n are allowed n cannot be 0, because this means no particle in the box

E= a2

h2

8 π2

m

a= nπL

En= n

2

h2

8mL2

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Particle in Box 2D

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Particle in Box 2D

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Particle in Box 3D

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Application of quantum mechanicsTranslational motionTranslational motion– The important concept in 2 and 3D box is The important concept in 2 and 3D box is degeneracydegeneracy

– 2 different wavefunction, same energy2 different wavefunction, same energy

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Application of quantum mechanicsTranslational motionTranslational motion– The important concept in 2 and 3D box is The important concept in 2 and 3D box is degeneracydegeneracy

– 2 different wavefunction, same energy2 different wavefunction, same energy

– Closely related to symmetryClosely related to symmetry

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Application of quantum mechanicsTranslational motionTranslational motion– TunnellingTunnelling

– Somehow, the particles can escape throughSomehow, the particles can escape through

classically forbidden regionclassically forbidden region

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Examples

Calculate the probability that a particle in the state with n = 1 will beCalculate the probability that a particle in the state with n = 1 will befound between x = 0.25L and x = 0.75L in a box of length L (with x = found between x = 0.25L and x = 0.75L in a box of length L (with x =

0 at the left-hand end of the box).0 at the left-hand end of the box).

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quiz

Calculate the probability that a particle will be found between 0.65LCalculate the probability that a particle will be found between 0.65Land 0.67L in a box of length L when it has (a) n = 1, (b) n = 2. Takeand 0.67L in a box of length L when it has (a) n = 1, (b) n = 2. Takethe wavefunction to be a constant in this range.the wavefunction to be a constant in this range.

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