The paper discusses the realization of the Quantum Spin Hall (QSH) effect in HgTe/CdTe semiconductor 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 identified as 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 use symmetry considerations and \( k \cdot p \) perturbation theory to show that the electronic states near the \(\Gamma\) point are described by a 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 presence of a gap closing transition at the \(\Gamma\) point in the HgTe/CdTe quantum wells confirms the theoretical prediction of the QSH state. The paper also discusses experimental methods for detecting the QSH effect, including electrical measurements and spin-filtered measurements, and highlights the potential for detecting the QSH state in existing HgTe/CdTe quantum well samples.The paper discusses the realization of the Quantum Spin Hall (QSH) effect in HgTe/CdTe semiconductor 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 identified as 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 use symmetry considerations and \( k \cdot p \) perturbation theory to show that the electronic states near the \(\Gamma\) point are described by a 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 presence of a gap closing transition at the \(\Gamma\) point in the HgTe/CdTe quantum wells confirms the theoretical prediction of the QSH state. The paper also discusses experimental methods for detecting the QSH effect, including electrical measurements and spin-filtered measurements, and highlights the potential for detecting the QSH state in existing HgTe/CdTe quantum well samples.