The quantitative condition has been widely used in the practical applications of the adiabatic theorem. However, it had never been proved to be sufficient or necessary before. It was only recently found that the quantitative condition is insufficient, but whether it is necessary remains unresolved. In this letter, we prove that the quantitative condition is necessary in guaranteeing the validity of the adiabatic approximation.PACS numbers: 03.65Ca, 03.65.Ta, 03.65.VfThe adiabatic theorem reads that if a quantum system with a time-dependent nondegenerate Hamiltonian H(t) is initially in the n-th instantaneous eigenstate of H(0), and if H(t) evolves slowly enough, then the state of the system at time t will remain in the n-th instantaneous eigenstate of H(t) up to a multiplicative phase factor. The theorem was first introduced eighty years ago [1], has been one of the most important theories in quantum mechanics [2][3][4][5][6] and has underpinned some of the most important developments in physical chemistry [7,8], quantum field theory [9], geometric phase [10], and quantum computing [11]. The practical applications of the theorem rely on the criterion of the "slowness" required by the theorem, which is usually encoded by the quantitative condition,where E m (t) and |E m (t) are the eigenvalues and eigenstates of H(t), and τ is the total evolution time. Although the sufficiency as well as necessity of the condition had been never proved before, it had been widely used as a criterion of the adiabatic approximation. It was only recently found that the quantitative condition is insufficient in guaranteeing the validity of the adiabatic approximation. Marzlin and Sanders [12] illustrated that perfunctory application of the adiabatic theorem may lead to an inconsistency. Tong et al [13] pointed out that the inconsistency is a reflection of the insufficiency of the adiabatic condition and they further showed that the condition cannot guarantee the validity of the adiabatic approximation. Indeed, for a given quantum system defined by Hamiltonian H a (t) with evolution operator U a (t) = T exp(−i t 0, one can always construct another quantum system defined by Hamiltonian H b (t) = iU † a (t)U a (t). The two systems fulfill the same adiabatic condition, but the adiabatic approxima- * Electronic address: tdm@sdu.edu.cn tion must be invalid for at least one of them, which indicates that the adiabatic condition is insufficient. These recent findings have stimulated a great number of reexaminations on the adiabatic approximation. Some papers contributed to the investigation of the reasons behind the insufficiency [14][15][16][17][18][19][20][21], while others contributed to the development of alternative conditions [22][23][24][25][26][27][28][29][30][31][32][33] or to the examination of the validity of the quantitative condition in concrete quantum systems [34][35][36][37][38][39][40]. However, so far, whether the quantitative condition is necessary remains unresolved. It is worth noting that some authors have claimed that th...

A 76-year-old woman with hypertension was in coma for 48 hrs, and its bed is exposed to anti-diabetic drugs, i.e., glibenclamide 100 tables/bottle leaving only 90 tables (2.5mg /piece). The patient had no exposure to carbon monoxide, and no history of diabetes mellitus and cardiac arrest. She used to live alone, the patient was unconscious for two days, only then discovered she fell ill, and then she was given 50 ml of sugar water oral. Unfortunately, the patient remained in a coma. Brain CT scans were normal. Blood pressure was 153/92mmHg.The respiratory rate and SpO2 were normal. She had a Glasgow coma score (GCS) of E1M4V1 with symmetry pupils (diameter 1.5mm), no light reflex, corneal reflex, and extraocular movements. On admission, laboratory blood glucose was 8.6 mmol/l. Electrolytes, liver and renal function was normal. ECG revealed normal. On the second day of hospital admission, brain DWI revealed diffuse high signals on the cerebral cortex bilaterally, hippocampus, and basal ganglia ( Figure 1 A and B). After four days of admission, the patient was still in deep coma and phlegm, so the tracheotomy breathing was performed. After five days, the patient was into a vegetative state, and she was discharged after 21 days. On follow-up 2 months later, the patient was a persistent vegetative state. Severe Hypoglycemic Coma Event on MRI: Specific Brain Necrosis