This paper presents a theoretical investigation of the Quantum Spin Hall (QSH) effect in HgTe/CdTe quantum wells. The authors show that the QSH effect, a topological state of matter distinct from conventional insulators, can be realized in these quantum wells. By varying the thickness of the quantum well, the electronic state transitions from a normal to an "inverted" type at a critical thickness $ d_c $. This transition is a topological quantum phase transition between a conventional insulating phase and a phase exhibiting the QSH effect with a single pair of helical edge states. The authors also discuss methods for experimental detection of the QSH effect. The QSH effect is a topological state of matter characterized by the presence of helical edge states, which are protected by time-reversal symmetry. These edge states are robust against impurity scattering and many-body interactions, and their existence is confirmed by numerical calculations. The QSH state is classified by a $ Z_2 $ topological invariant, which distinguishes it from conventional insulators. The integer quantum Hall effect is also a topological state of matter, characterized by a topological integer $ n $, which determines the quantized value of the Hall conductance and the number of chiral edge states. The authors show that the QSH state can be realized in the "inverted" regime of HgTe/CdTe quantum wells, where the well thickness $ d $ is greater than a critical thickness $ d_c $. In this regime, the electronic states near the $ \Gamma $ point are described by the relativistic Dirac equation in 2+1 dimensions. At the quantum phase transition at $ d = d_c $, the mass term in the Dirac equation changes sign, leading to two distinct $ U(1) $-spin and $ Z_2 $ topological numbers on either side of the transition. The stability of the helical edge states is confirmed by numerical calculations, and the time-reversal property leads to the $ Z_2 $ classification of the QSH state. The authors also discuss the experimental detection of the QSH effect. They propose that a series of purely electrical measurements can be used to detect the basic signature of the QSH state. By sweeping the gate voltage, one can measure the two-terminal conductance $ G_{LR} $ from the p-doped to bulk-insulating to n-doped regime. In the bulk-insulating regime, $ G_{LR} $ should vanish at low temperatures for a normal insulator at $ d < d_c $, while $ G_{LR} $ should approach a value close to $ 2e^2/h $ for $ d > d_c $. In a six-terminal measurement, the QSH state would exhibit vanishing electric voltage drop between the terminals $ \mu_1 $ and $ \mu_2 $, and between $ \mu_3 $ and $ \muThis paper presents a theoretical investigation of the Quantum Spin Hall (QSH) effect in HgTe/CdTe quantum wells. The authors show that the QSH effect, a topological state of matter distinct from conventional insulators, can be realized in these quantum wells. By varying the thickness of the quantum well, the electronic state transitions from a normal to an "inverted" type at a critical thickness $ d_c $. This transition is a topological quantum phase transition between a conventional insulating phase and a phase exhibiting the QSH effect with a single pair of helical edge states. The authors also discuss methods for experimental detection of the QSH effect. The QSH effect is a topological state of matter characterized by the presence of helical edge states, which are protected by time-reversal symmetry. These edge states are robust against impurity scattering and many-body interactions, and their existence is confirmed by numerical calculations. The QSH state is classified by a $ Z_2 $ topological invariant, which distinguishes it from conventional insulators. The integer quantum Hall effect is also a topological state of matter, characterized by a topological integer $ n $, which determines the quantized value of the Hall conductance and the number of chiral edge states. The authors show that the QSH state can be realized in the "inverted" regime of HgTe/CdTe quantum wells, where the well thickness $ d $ is greater than a critical thickness $ d_c $. In this regime, the electronic states near the $ \Gamma $ point are described by the relativistic Dirac equation in 2+1 dimensions. At the quantum phase transition at $ d = d_c $, the mass term in the Dirac equation changes sign, leading to two distinct $ U(1) $-spin and $ Z_2 $ topological numbers on either side of the transition. The stability of the helical edge states is confirmed by numerical calculations, and the time-reversal property leads to the $ Z_2 $ classification of the QSH state. The authors also discuss the experimental detection of the QSH effect. They propose that a series of purely electrical measurements can be used to detect the basic signature of the QSH state. By sweeping the gate voltage, one can measure the two-terminal conductance $ G_{LR} $ from the p-doped to bulk-insulating to n-doped regime. In the bulk-insulating regime, $ G_{LR} $ should vanish at low temperatures for a normal insulator at $ d < d_c $, while $ G_{LR} $ should approach a value close to $ 2e^2/h $ for $ d > d_c $. In a six-terminal measurement, the QSH state would exhibit vanishing electric voltage drop between the terminals $ \mu_1 $ and $ \mu_2 $, and between $ \mu_3 $ and $ \mu