Cơ học lượng tử 1.3

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Outline

Solving the Simple Harmonic Oscillator with the ladder operators
Representing an operator as a matrix
Heisenberg Picture and Schroedinger Picture
Equations of motion for x(t) and p(t) in the Heisenberg Picture
The Ehrenfest Theorem

Please read Goswami Chapter 8
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Up to now we have viewed everything from the Schroedinger perspective (that is, the Schroedinger Equation is a time-development equation for Ψ.
Now consider the Heisenberg Picture and find a time-development equation for A’.

Start with the definition:
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Outline

The WKB Approximation: Introduction
WKB Connection Formulas
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The WKB Approximation: Introduction

The issue: Most potentials in real applications are not simple square wells and so forth, so generally they lead to differential equations that are hard to solve.
Generally solving these requires making approximations.
There is an approximation that works well if V varies only slowly as a function of x, so if we look in a small region, we can say that V~ constant. This is the WKB Approximation.
The method:
Consider a confining potential that is generally arbitrarily shaped but that does not vary rapidly:






Consider a particle trapped in the well at E.
Definition: The values of x for which V=E are called the “turning points.”
V(x)

E
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Write down the Schroedinger Equation, assume that because V is ~ constant in a local region, ψ is ~ a free particle in that region: that is, a plane wave. Thus assume that ψ ~ Aeikx.
Plane waves do not change their amplitudes, so assume that δ2A/dx2=0.
Solve the Schroedinger Equation with this approximation.
The approximate solution is close to the exact solution everywhere except at the turning points.
(3) To repair the problem at the turning points:
in those regions only, assume V is a linear function for which the Schroedinger Equation is easily solved.





Find ψ for that V at those x’s.
Connect the ψ’s at the turning points to the ψ’s that are everywhere else.
This is the boundary condition application. This develops equations called the Connection Formulas.
(5) The formulas for ψ’s that are produced by this method are general enough to be used in all problems where V is slowly varying.
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Reg 1 Reg 2 Reg 3
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Range 3
Range 4
Range 2 (approach righthand turning point from Region 2
Range 1 (approach right hand turning point from Region 1
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E
x =0
V(x)
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x1
x2
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Outline

Systems with 2 degrees of freedom: Introduction
Exchange Degeneracy
The Exchange Operator

Please read Goswami Chapter 9.
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Systems with 2 degrees of freedom: Introduction

Examples of kinds of degrees of freedom:
2 particles free to move in 1 dimension
1 particle free to move in 2 dimensions

Each of these leads to energy degeneracy.
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Outline

System of 2 interacting particles in 1 dimension
System of 1 particle in 2 dimensions
Multi (>2) particle systems in 3 dimensions

Please read Goswami Chapter 11.
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Conclusions about this:

The X equation concerns the motion of the center of mass. Note that there is no V acting on the center of mass.

The x equation concerns the motion of the reduced mass (this is mathematically equivalent to a body of finite mass orbiting in the V of an immobile, infinitely massive other body. Since the reduced mass does respond to the V, the V is in that equation.

When the Schroedinger Equation is expressed in terms of u(x)U(X), the motion of M and μ are decoupled, independent. But when the Schroedinger Equation is expressed in terms of (x1, x2), the behaviors of the real physical particles (m1, m2) cannot be decoupled. They remain really physically correlated, even when separated by great distances. This implies a philosophical question: are the 2 particles truly correlated---for example, does measuring the position of m1 disrupt the momentum of m2? This is the Einstein-Podolsky-Rosen (EPR) Paradox.
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Outline

Angular momentum introduction
Angular momentum commutators
Representing the L operators and the |λ,m’> wavefunctions in r-θ-ϕ space.
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Angular momentum introduction

Why is this important?
Any physical system that has rotational motion has energy associated with that motion. That rotation must somehow be reflected in the Hamiltonian in order to correctly and fully describe the system’s energy (which is quantized by it). The rotation is also reflected in the ψ, so the rotational status is input to the system’s characteristic as ψ(symmetric) or ψ(antisymmetric). Thus the rotational behavior influences the system’s response to the Pauli Exclusion Principle.
2. This gives us a motivation to discuss how to invent a Hamiltonian. Whenever possible, people create quantum mechanical Hamiltonians by writing down the classical Hamiltonian for a system and then calling everything but known constants operators.

How to find the quantum mechanical Hamiltonian for a particle that is orbiting at a constant radius R about a point in 3-dimensions.
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Outline

Graphical representation of angular momentum
Spherical harmonics
The rigid rotator
Generalized angular momentum
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Outline

Angular momentum ladder operators
Finding mJ and λJ
Normalizing the |λJmJ>
Lz is the generator of rotations
Conservation of angular momentum in quantum mechanics
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