KI2141-2015 SIK Lecture02c RotationMotion

Quantum Theory : Techniques & ApplicationsRotational Motion

Achmad Rochliadi, Ph.D. Program Studi Kimia

Institut Teknologi Bandung

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Rotational motionRotational motion

The rotational motion of a particle about a central point is described by its angular momentum, J. The angular momentum is a vector: its magnitude gives the rate atwhich a particle circulates and its direction indicates the axis of rotation.

Momen Inertia

Angular velocity

Torque, twisting force, force to accelerate a rotation

Kinetic energy increase due to torque for τ second

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Rotational in 2 dimensionRotational in 2 dimension

A particle of mass m constrained to move in a circular path of radius r in the xy-plane with constant Potential Energy

V = 0 , the Total Energy is equal Kinetic Energy

The angular momentum, Jz,

The Energy become

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Qualitative origin of quantized rotationQualitative origin of quantized rotation

We have that rotational momentum and de Broglie relation,

rotational momentum become

AcceptableNot acceptable

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Allowed wavelengthAllowed wavelength

Energy of the particles have to be quantized, the allowed wavelength are

Allowed rotational momentum,

Energy level of particles on a ring

Angular momentum of a particles on a ring

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The normalized wavefunctionThe normalized wavefunction

The normalized general solution ..

Hamiltonian for particle

The radius is fixed, Hamiltonian, and Scrodinger equation becomes

Using cylindrical coordinate

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Acceptable solutionAcceptable solution

The normalized general solution ..

Wavefunction must single-valued → ψ must satisfy cyclic boundary condition

The solution :

Due to So :

Requirement

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Representation of the wave functionRepresentation of the wave function

The real part of the wave function of a particle on a ring

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Rotation in three dimentionRotation in three dimention

In three dimention → A sphere

(a) The wavefunction of a particle on a spherical surface must satisfy simultaneouslytwo cyclic boundary conditions. (b) The energy and angular momentum of a particle on a sphere are quantized.

(c) Space quantization is the restriction of the component of angular momentumaround an axis to discrete values.

(d) The vector model of angular momentum uses diagrams to represent the state of angular momentum of a rotating particle

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The Schrodinger equationThe Schrodinger equation

The hamiltonian for rotation motion in 3 dimentions,

The Schrodiner equation became

Radius, r, constant, and using the separation of variable methods, the wave function became

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The Schrodinger equationThe Schrodinger equation

The laplacian, and legendrian in spherical polar coordinate is

The Schrodinger equation become

And with

The Schrodinger also

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Separation methodsSeparation methods

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Separation methodsSeparation methods

The separated equation become.

The cyclic boundary conditions on Θ arising from the need for the wavefunctions to match at θ=0 and 2π (the North Pole) result in the introduction of a second quantum number, l.

The presence of the quantum number ml in the second

equation implies, as we see below, that the range of acceptable values of m

l is restricted by the value of l

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The wave functionsThe wave functions

The normalize wavefunction

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The wave functions representationThe wave functions representation

Representation of the wave function

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Space quantitationSpace quantitation

Permitted orientation of angular momentum when l = 2

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SpinSpin

● Spin is an intrinsic angular momentum of a fundamental particle.

● A fermion is a particle with a half-integral spin quantum number.

● A boson is a particle with an integral spinquantum number.

● For an electron, the spin quantum number is s= 1/2 . The spin magnetic quantum number is m

s=s,s−1,..., −s; for an electron, m

s=± 1/2.

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SpinSpin

Electron spin have two orientation

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SpinSpin

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Thank You

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