Klein悖論是物理學家Oskar Klein在西元1929年間，利用Dirac方程式，分析處於強電場中的電子之散射現象時，發現一個意外的結果。他對該電子進行分析，後來發現到電子的反射係數會大於一，穿透係數會小於零。以相對論性量子力學的理論來看待這個結果，是令人難以理解的，原因是當時的物理學家對Dirac方程式的詮釋是錯誤的，後來才發現這個意外的結果其實是跟粒子的生成與湮滅有關。他們發現相對論量子力學的理論不夠完整，並以新的概念來對Dirac方程式作詮釋，後來漸漸建立了量子場論這套新的理論。
在這篇論文中，首先會介紹相對論量子力學的發展史，第二～三章會介紹相對論性自由粒子的負能量解及介紹Klein-Gordon方程式及Dirac方程式的代數性質，第四～六章則會利用Schrödinger方程式，Pauli方程式，Klein-Gordon方程式以及Dirac方程式，來對Klein悖論做延伸性的思考，來說明為何相對論量子力學的理論架構不夠完整。

Klein Paradox was an unexpected result published by Oskar Klein in 1929. He obtained a surprising result by applying the Dirac equation to study the electron scattering off a strong potential. Klein assumed that the potential energy of the electron was a high step function potential and found that the transmission coefficient was greater than unity and the reflection coefficient was negative. This result cannot be explained within the framework of relativistic quantum mechanics. It is known that the paradox can be resolved by considering quantum field theory in which pair production and annihilation of the particles and anti-particles are essential properties. 　In this thesis, we shall describe the Klein paradox in relativistic quantum mechanics. We shall explicitly calculate the solutions of the Schrödinger equation, the Klein-Gordon equation and the Dirac equation of a particle in several potentials, which include the step function potential and the potential barrier. The calculations of the solutions of the equations with the potentials are given. We make no attempt to resolve the paradox by considering quantum field theory.

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