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Quantum state
In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system represented by the state. Knowledge of the quantum state, and the rules for the system's evolution in time, exhausts all that can be known about a quantum system.
Quantum states may be defined differently for different kinds of systems or problems. Two broad categories are
Historical, educational, and application-focused problems typically feature wave functions; modern professional physics uses the abstract vector states. In both categories, quantum states divide into pure versus mixed states, or into coherent states and incoherent states. Categories with special properties include stationary states for time independence and quantum vacuum states in quantum field theory.
As a tool for physics, quantum states grew out of states in classical mechanics. A classical dynamical state consists of a set of dynamical variables with well-defined real values at each instant of time. For example, the state of a cannon ball would consist of its position and velocity. The state values evolve under equations of motion and thus remain strictly determined. If we know the position of a cannon and the exit velocity of its projectiles, then we can use equations containing the force of gravity to predict the trajectory of a cannon ball precisely.
Similarly, quantum states consist of sets of dynamical variables that evolve under equations of motion. However, the values derived from quantum states are complex numbers, quantized, limited by uncertainty relations, and only provide a probability distribution for the outcomes for a system. These constraints alter the nature of quantum dynamic variables. For example, the quantum state of an electron in a double-slit experiment would consist of complex values over the detection region and, when squared, only predict the probability distribution of electron counts across the detector.
The process of describing a quantum system with quantum mechanics begins with identifying a set of variables defining the quantum state of the system. The set will contain compatible and incompatible variables. Simultaneous measurement of a complete set of compatible variables prepares the system in a unique state. The state then evolves deterministically according to the equations of motion. Subsequent measurement of the state produces a sample from a probability distribution predicted by the quantum mechanical operator corresponding to the measurement.
The fundamentally statistical or probabilisitic nature of quantum measurements changes the role of quantum states in quantum mechanics compared to classical states in classical mechanics. In classical mechanics, the initial state of one or more bodies is measured; the state evolves according to the equations of motion; measurements of the final state are compared to predictions. In quantum mechanics, ensembles of identically prepared quantum states evolve according to the equations of motion and many repeated measurements are compared to predicted probability distributions.
Measurements, macroscopic operations on quantum states, filter the state. Whatever the input quantum state might be, repeated identical measurements give consistent values. For this reason, measurements 'prepare' quantum states for experiments, placing the system in a partially defined state. Subsequent measurements may either further prepare the system – these are compatible measurements – or it may alter the state, redefining it – these are called incompatible or complementary measurements. For example, we may measure the momentum of a state along the axis any number of times and get the same result, but if we measure the position after once measuring the momentum, subsequent measurements of momentum are changed. The quantum state appears unavoidably altered by incompatible measurements. This is known as the uncertainty principle.
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Quantum state
In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system represented by the state. Knowledge of the quantum state, and the rules for the system's evolution in time, exhausts all that can be known about a quantum system.
Quantum states may be defined differently for different kinds of systems or problems. Two broad categories are
Historical, educational, and application-focused problems typically feature wave functions; modern professional physics uses the abstract vector states. In both categories, quantum states divide into pure versus mixed states, or into coherent states and incoherent states. Categories with special properties include stationary states for time independence and quantum vacuum states in quantum field theory.
As a tool for physics, quantum states grew out of states in classical mechanics. A classical dynamical state consists of a set of dynamical variables with well-defined real values at each instant of time. For example, the state of a cannon ball would consist of its position and velocity. The state values evolve under equations of motion and thus remain strictly determined. If we know the position of a cannon and the exit velocity of its projectiles, then we can use equations containing the force of gravity to predict the trajectory of a cannon ball precisely.
Similarly, quantum states consist of sets of dynamical variables that evolve under equations of motion. However, the values derived from quantum states are complex numbers, quantized, limited by uncertainty relations, and only provide a probability distribution for the outcomes for a system. These constraints alter the nature of quantum dynamic variables. For example, the quantum state of an electron in a double-slit experiment would consist of complex values over the detection region and, when squared, only predict the probability distribution of electron counts across the detector.
The process of describing a quantum system with quantum mechanics begins with identifying a set of variables defining the quantum state of the system. The set will contain compatible and incompatible variables. Simultaneous measurement of a complete set of compatible variables prepares the system in a unique state. The state then evolves deterministically according to the equations of motion. Subsequent measurement of the state produces a sample from a probability distribution predicted by the quantum mechanical operator corresponding to the measurement.
The fundamentally statistical or probabilisitic nature of quantum measurements changes the role of quantum states in quantum mechanics compared to classical states in classical mechanics. In classical mechanics, the initial state of one or more bodies is measured; the state evolves according to the equations of motion; measurements of the final state are compared to predictions. In quantum mechanics, ensembles of identically prepared quantum states evolve according to the equations of motion and many repeated measurements are compared to predicted probability distributions.
Measurements, macroscopic operations on quantum states, filter the state. Whatever the input quantum state might be, repeated identical measurements give consistent values. For this reason, measurements 'prepare' quantum states for experiments, placing the system in a partially defined state. Subsequent measurements may either further prepare the system – these are compatible measurements – or it may alter the state, redefining it – these are called incompatible or complementary measurements. For example, we may measure the momentum of a state along the axis any number of times and get the same result, but if we measure the position after once measuring the momentum, subsequent measurements of momentum are changed. The quantum state appears unavoidably altered by incompatible measurements. This is known as the uncertainty principle.