Mental illness remains one of the most critical public health issues. Despite its importance, many mental health professionals highlight a disconnect between their training and actual real-world patient practice. To help bridge this gap, we propose PATIENT-Ψ, a novel patient simulation framework for cognitive behavior therapy (CBT) training. To build PATIENT-Ψ, we construct diverse patient cognitive models based on CBT principles and use large language models (LLMs) programmed with these cognitive models to act as a simulated therapy patient. We propose an interactive training scheme, PATIENT-Ψ-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-Ψ. To evaluate PATIENT-Ψ, we conducted a comprehensive user study of 13 mental health trainees and 20 experts. The results demonstrate that practice using PATIENT-Ψ-TRAINER 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-Ψ is perceived to be closer to real patient interactions than GPT-4, and PATIENT-Ψ-TRAINER holds strong promise to improve trainee competencies. We will release all our code and data upon acceptance of this paper.