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Title Page

ABSTRACT

Contents

Chapter 1. Introduction 10

Chapter 2. Quantum Mechanical Correspondence of Linear Transformations for General Hamiltonian Systems 16

2.1. Classical Linear Canonical Transformation between Two Systems 16

2.2. Quantum Mechanical Counterpart for the Linear Canonical Transformation 22

2.3. Quantum Operators in each space and their Average Values 29

2.4. Special Types for Linear Canonical Transformations 35

2.4.1. Scale Transformation 35

2.4.2. Gauge Transformation 42

Chapter 3. Operator Ordering 46

Chapter 4. Linear Transformations of 2-Dimensional Hamiltonian Systems 52

4.1. Linear Canonical Transformation 52

4.2. Unitary Transformation 56

4.3. Example - Coupled Forced Damped Harmonic Oscillator 60

Chapter 5. Linear Transformations on N-Dimensional Hamiltonian Systems 72

5.1. Linear Canonical Transformation 72

5.2. Unitary Transformation 75

5.3. Examples 78

5.3.1. Coupled Forced Damped Harmonic Oscillator 78

5.3.2. 3-Coupled Damped Harmonic Oscillator 80

5.3.3. 2-Dimensional Rotating Coordinate System 83

5.3.4. 3-Dimensional Rotating Coordinate System 89

Chapter 6. Conclusions and Further Studies 92

Bibliography 95

List of Figures

Figure 2.1. A typical representation of a path x(t) 23

Figure 2.2. Wave function of a normalized damped harmonic oscillator for the quantum number 0, β＝0.1. 37

Figure 2.3. Wave function of a normalized damped harmonic oscillator for the quantum numbers 1, β＝0.1. 38

Figure 2.4. Wave function of a normalized damped harmonic oscillator for the quantum number 2, β＝0.1. 39

Figure 4.1. Wave function of a normalized 2-dimensional damped harmonic-oscillator for the quantum numbers 1, 1, where v＝0.1, α₁,₂ ＝0, t＝0. 68

Figure 4.2. Wave function of a normalized 2-dimensional damped harmonic-oscillator for the quantum numbers 1, 1, where v＝0.1, α₁,₂ ＝0, t＝20. 69

Figure 4.3. Wave function of a normalized 2-dimensional damped harmonic-oscillator for the quantum numbers 1, 1, where v＝0.1, α₁,₂ ＝0, t＝40. 70

Figure 4.4. Wave function of a normalized 2-dimensional damped harmonic-oscillator for the quantum numbers 1, 1, where v＝0.1, α₁,₂ ＝0, t＝50. 71

Figure 5.1. Wave function of a normalized harmonic-oscillator of the rotating coordinate system for the quantum numbers 1, 1, where φ＝π/4.[이미지참조] 86

Figure 5.2. Wave function of a normalized harmonic-oscillator of rotating coordinate system for the quantum numbers 1, 1, where φ＝π/2.[이미지참조] 87

Figure 5.3. Wave function of a normalized harmonic-oscillator of rotating coordinate system for the quantum numbers 1, 1, where φ＝3π/4.[이미지참조] 88

초록보기

 Classical Hamiltonians help us to deal with a system quantum mechanically as well as to get the classical equation of motion. In a general N-dimensional coordinate system, for a general classical Hamiltonian, a canonical transformation is sought, and a new classical Hamiltonian corresponding to the old Hamiltonian is found, where a linear transformation is used for convenience.

For a given Hamiltonian, while the classical equation of motion is easily obtained, the identification of the corresponding quantum mechanical Hamiltonian is not so easy. There are ordering problems in the quantum mechanical operators, which can be solved by using two methods separately. The one is to use Feynman path integral which comes from the principle that the quantum mechanical paths of the motion of a particle between two points are all the paths between the two points, while the classical path is the one that makes the action be minimum. The second method is α-ordering in which the completeness and non-commutativity in quantum mechanics is used for a given quantum Hamiltonian and a given Hilbert space. Even though the starting point of the two methods are not same, the same results are obtained. Feynman path integral method does not show the exact results for the case when the order of the momentum is greater than 2. However, it gives exact results for any order in coordinate.

A unitary operator corresponding to a classical canonical transformation is introduced. The form of the unitary operator is not unique, and the difficulty of the calculation depends on the choice of a unitary operator. Using the unitary transformation, as an example, we obtain a solution of a time-dependent system with a known quantum mechanical solution. The uncertainty relation between the coordinate and canonical momentum, and that between the coordinate and mechanical momentum is also obtained.

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