Dimensionality and vectors in quantum mechanics

[chris_183]

i was reading about the mathematical formulation of quantum mechanics and how a system at any given time is described by a vector represented by an infinite number of spatial complex number coordinates. does this infinite-dimensional state space have any physical significance or is it just a mathematical abstraction?

 

[Fredrik]

A vector in that space represents the state of a physical system. It's time evolution is given by the Schrödinger equation.

In case you need something to compare that to: In classical mechanics the state is given by a specification of the location and velocity of all component parts, and the time evolution is given by Newton's second law.

If you assume that the state vector is an accurate description of reality instead of just a mathematical tool (needed to calculate the probabilities of the possible results of an experiment), what you get is the many-worlds interpretation of quantum mechanics. So is that assumption/interpretation correct or not? The question is actually irrelevant. It's kind of like asking if these three dots (...) are really two dots to the left of one dot, or one dot to the left of two dots.

 

[denian]

The concept of dimensionality and vectors in quantum mechanics is a fundamental aspect of the mathematical framework used to describe quantum systems. While it may seem like an abstract concept, it has significant physical significance in understanding the behavior and properties of particles at the quantum level.

The infinite-dimensional state space is not just a mathematical abstraction, but it represents the full range of possible states that a quantum system can exist in. These states are represented by vectors, which contain information about the position, momentum, and other physical properties of the particles. The complex number coordinates in the vector represent the probability amplitudes of each possible state, and when combined, they give us a complete picture of the system.

Furthermore, the concept of dimensionality in quantum mechanics goes beyond just the spatial dimensions we are familiar with in classical physics. It also includes additional dimensions, such as spin, which plays a crucial role in determining the behavior of particles. This multidimensional state space allows for a more comprehensive and accurate description of the quantum world, and has been validated through numerous experiments and observations.

In summary, the infinite-dimensional state space and vectors in quantum mechanics are not just mathematical abstractions, but they have physical significance in accurately describing the behavior and properties of particles at the quantum level. They play a crucial role in our understanding of the quantum world and have been essential in the development of many technological advancements.