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MHD93
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I wonder and feel like knowing how the product of psi with its complex conjugate represents the Probability Density for the object to be there? how two (apparent) different concepts are linked that way?
Essentially because the product of a complex number by its coniugate is the square modulus of the complex number.Mohammad_93 said:I wonder and feel like knowing how the product of psi with its complex conjugate represents the Probability Density for the object to be there? how two (apparent) different concepts are linked that way?
Mohammad_93 said:but what makes it concern probability?
Mohammad_93 said:I wonder and feel like knowing how the product of psi with its complex conjugate represents the Probability Density for the object to be there? how two (apparent) different concepts are linked that way?
The requirement that the trace is equal to 1 is the assumption of conservation of probability, so there is already an implicit assumption of the probabilistic interpretation in that statement.A. Neumaier said:In particular, the diagonal elements p_k:= rho_{kk} are nonnegative and satisfy sum p_k = Tr rho = 1. Thus they look like probabilities. Observables are represented by arbitrary Hermitian matrices X, and their expectation in the state rho is defined to be <X>= Tr rho X.
bobbytkc said:The requirement that the trace is equal to 1 is the assumption of conservation of probability, so there is already an implicit assumption of the probabilistic interpretation in that statement.
alxm said:If you totally disregard the Born rule and assume nothing about probabilities and measurements, you still have the Schrödinger equation giving a set of energy eigenstates for the system and their time evolution.
Normalization and its conservation is then required for conservation of energy.
A. Neumaier said:Thus normalization is optional, and a change of normalization is just like a change of global phase.
In Hamiltonian quantum mechanics, yes. My point was that conservation f normalization is a consequence of the Schroedinger dynamics, and must not be _required_, as you had formulated it.alxm said:Sure, but two different wave functions describing physically-identical systems should reasonably be expected to have the same observables.
So while the normalization is arbitrary, it only really makes sense to have it set to 1.
"Psi" is a variable used in probability theory to represent the likelihood or probability of a certain event occurring. It is often denoted by the Greek letter "Ψ" and is commonly used in complex mathematical equations to calculate the probability of an event.
"Psi" is often used in conjunction with other concepts in probability theory, such as random variables, probability distributions, and conditional probabilities. It helps to quantify the likelihood of certain outcomes and is an important tool in analyzing complex systems.
Yes, "Psi" can be applied to real-life situations in many fields, such as finance, physics, and biology. It is used to model and predict the outcomes of complex systems, and has practical applications in risk analysis, decision making, and scientific research.
"Psi" is a variable used in probability theory, while probability refers to the likelihood or chance of an event occurring. "Psi" is a tool used to calculate probability, but it is not the same as probability itself.
"Psi" and probability are essential in understanding complex concepts by providing a mathematical framework for analyzing and predicting the outcomes of complex systems. They allow us to quantify uncertainty and make informed decisions based on statistical analysis.