In quantum physics, the dimension of a state vector is determined by the number of independent variables required to specify the state of a quantum system. The state vector, also known as a ket vector, is an element of a complex vector space called a Hilbert space.
The dimension of the state vector depends on the number of possible outcomes or states that the system can have. For example, if a system has two possible outcomes, such as a spin-1/2 particle with spin-up and spin-down states, the dimension of the state vector would be 2. In this case, the state vector can be represented as a 2-dimensional column vector.
In general, if a system has N possible outcomes or states, the dimension of the state vector would be N. The state vector is often represented as an N-dimensional column vector, where each element corresponds to the probability amplitude or complex coefficient associated with a particular state.
It's important to note that the dimension of the state vector can change depending on the properties of the quantum system. For example, a system with multiple particles, each having their own independent states, would have a higher-dimensional state vector that combines the individual state vectors of each particle.
The dimension of the state vector is a fundamental aspect of quantum mechanics, as it determines the mathematical space in which quantum states and operations are described.