1. Ch 2. The Schrödinger Equation (S.E)
- Due to the contribution of wave-particle duality, an
appropriate wave equation need to be solved for the microscopic world.
Erwin Schrödinger was the first to formulate such an equation
- We need to be familiar with operators, eigenfunction, wavefunction, eigenvalues that are used in S.E.
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2.1 What determines if a system needs to be described using Q.M?
When do we use a particle description(classical) of an atomic or
molecular system and when do we use a wave (quantum mechanical) description?
two criteria are used!
1) The magnitude of the wavelength of the particle relative to the
dimension of the problem, a hydrogen molecule 100 pm vs.
a baseball 1x m
no way to show wave character of baseball
2) The degree to which the allowed energy values form a continu
ous energy spectrum
Boltzmann distribution :
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There are two limits
1) large T or small εi – εj
: small energy gap(almost continuous energy),
classical behavior
2) (εi – εj)/kT >> 1
: large energy gap, quantum behavior
Also we can derive the internal energy by Boltzmann distribution
k : the Boltzmann constant
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Relative population in the different energy levels
a) Sharp energy levels
b) ∆E ≈ kT : classical behavior(nearby the continuous energy)
c) ∆E >> kT : quantum behavior(discrete energy)
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2.2 Classical waves and the nondispersive wave equation
Example of waves
a) Plane waves
b) spherical waves
c) cylindrical waves
Wave front : surface over the maximum or minimum amplitude
Direction of propagation of waves : blue arrows, perpendicular
to the surface
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Mathematically, we can describe the wave by wave function.
Amplitude of wave : related to position and time
Position : propagation depends on wavelength λ
Time : propagation depends on period T
Arbitrary condition : Ψ(0,0)=0
Positions where amplitude is zero :
x or t increase →
wave moves in the positive x direction
Graph of wave functions
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We use another form, too.
k : wave vector,
ω : angular frequency, ω=2πν
Choice of a zero in time or position : arbitrary(free)
→ We can rewrite the wave equation
In this case, Ψ(0,0) ≠ 0 and φ : initial phase
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If two or more waves in same region : interference
1) constructive interference : enhancement of amplitude
2) destructive interference : cancellation of amplitude
Phase difference in interference
1) constructive interference : 2nπ → same sign
2) destructive interference : (2n+1)π → opposite sign
Also, we can think about two waves same frequency,
amplitude and opposite directions
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Use this formula
Final result is simple than original equation
position of nodes are same at all time!
→ ‘standing’ waves
standing waves represent the stationary state(no change by time)
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Both of traveling waves and stationary waves, amplitude
and distance is not independent
: classical nondispersive wave eq.
v : velocity of wave propagation
This equation is a start point of the Schrödinger Equation.
Ex) 2-2. show that traveling wave satisfies wave equation.
Solution)
Equating two results, v=ω/k
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2.3 Waves represented by complex functions
We know the traveling wave equation
Where φ’ = φ - π/2
Use the Euler’s formula
We write the wave equation in complex form
Why uses a complex form?
1) All quantities can obtain by this form
2) easier than real form with differentiation and integration.
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Important formulas
1) Complex number : a+ib(a,b:real)
or equivalent form,
2) Complex conjugate of f(number of function) : f*
substituting i to –i
ex) (a+ib)* = a-ib,
3) Magnitude of f(number of function) : |f|
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2.4 The Schrödinger Equation
How we can obtain the Schrödinger Equation?
→ start with the classical wave equation and stationary wave
Substitute the wave equation by stationary wave, we obtain
Using the relations ω=2πν and νλ=v
Until now, it is classical wave.
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‘Introduce’ the quantum mechanics by de Broglie relation
Momentum is related by total energy E and potential energy V(x)
Substituting the wave equation by this relation
Use ℏ=h/2π, we obtain the time-independent Schrödinger Eq.
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How one obtain the time-dependent Schrödinger Equation?
Starting from the solution of classical wave equation
On the other hand
Then
Wave equivalent of a free particle of
energy E and momentum p moving on the x-direction
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Let’s try now,
We know that
Now replace
Time-dependent
Schrödinger Eq.
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One of our focus is stationary system.
In this case, both of time-dependent and time-independent are satisfied.
For stationary state,
Substitute the equation, we obtain
Finally,
→ same form as classical standing wave
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2.5 Solving the Schrödinger Equation
Key concept : operators, observables, eigenfunctions and eigenvalues
Operators in a classical mechanics
Ex) Velocity in Newton’s second law
1) Integrate the force acting on the particle over the interval
2) Multiply by the inverse of the mass
3) Add the quantity to the velocity at time t1
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How about a operators in Quantum Mechanics?
→ every measurable quantities(observables) have their operator each.
(ex : energy, momentum, position)
Notation : caret, Ô
Differential equations : set of solutions
Operator Ô has a set of eigenfunctions and eigenvalues
Ψn : eigenfunctions, an : eigenvalues
Ex) hydrogen atom
eigenfunctions : each orbitals(1s, 2s, 2px, …)
eigenvalues : each orbiral energies
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We see the time-independent Schrödinger equation.
This equation can be written by
Important physical implication : measurement process in Q.M.
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2.6 Eigenfunctions of Q.M. operator are orthogonal
Orthogonality in 3-dimensional vector space :
x•y = x•z = y•z = 0
Similar, orthogonality in functional space is defined by
If I = j, the integral has a nonzero value
Functions can be normalized and form an orthonormal set.
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In 3-dimension, normalization must be 3-dimension.
Closed-shell atoms are spherical symmetric and we normalized the wave functions in spherical coordinate
Volume element in spherical coordinate : r2 sin θ dr dθ dφ, not a dr dθ dφ
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2.7 Eigenfunctions of Q.M. operator form a complete set
completeness in 3-dimensional vector space :
Any vector in 3-dimensional can be represented by linear combination of vector x, y, and z
Similar, completeness in functional space :
Wave function can be expanded in the eigenfunctions of any Q.M. operator
It is same formalism as a Fourier series.
We choose a periodic function in [-b,b]
Fourier series is represented by
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If f(x) even : cn=0, dn calculated by orthogonality
Generally
And this approximation is nearly exact.
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Accuracy of Fourier series
a) Yellow line : real function
b) red line : fourier series approximation, n=2,3,4 and 6
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Summary
The time-dependent and time-independent Schrödinger equations play the role in solving quantum mechanical problems.
Operators, eigenfunctions, wave functions and eigenvalues are key concepts to solve quantum mechanical wave equations.
MS310 Quantum Physical Chemistry