Mental illness remains one of the most critical public health issues, with a significant gap between the available mental health support and patient needs. Many mental health professionals highlight a disconnect between their training and real-world patient interactions, leaving some trainees feeling unprepared and potentially affecting their early career success. In this paper, we propose PATIENT-{\Psi}, a novel patient simulation framework for cognitive behavior therapy (CBT) training. To build PATIENT-{\Psi}, we constructed diverse patient profiles and their corresponding cognitive models based on CBT principles, and then used large language models (LLMs) programmed with the patient cognitive models to act as a simulated therapy patient. We propose an interactive training scheme, PATIENT-{\Psi}-TRAINER, for mental health trainees to practice a key skill in CBT -- formulating the cognitive model of the patient -- through role-playing a therapy session with PATIENT-{\Psi}. To evaluate PATIENT-{\Psi}, we conducted a user study of 4 mental health trainees and 10 experts. The results demonstrate that practice using PATIENT-{\Psi}-TRAINER greatly enhances the perceived skill acquisition and confidence of the trainees beyond existing forms of training such as textbooks, videos, and role-play with non-patients. Based on the experts' perceptions, PATIENT-{\Psi} is perceived to be closer to real patient interactions than GPT-4, and PATIENT-{\Psi}-TRAINER holds strong promise to improve trainee competencies. Our pioneering patient simulation training framework, using LLMs, holds great potential to enhance and advance mental health training, ultimately leading to improved patient care and outcomes. We will release all our data, code, and the training platform.