ki2141-2015 sik lecture02c rotationmotion
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Rotational MottionsTRANSCRIPT
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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