Quantum States Part 2

Last time we had basic introduction about the quantum states ,This time we will study those in a brief . As we all know there are two types of quantum states PURE state and MIXED state , The pure state describes a position (spin) of a subatomic particle by a single KET vector while a mixed state is a mixture of pure states.

PURE STATE:

In last post we have seen that the basic unit of the data in fundamental computing is bit in which the maximum states can be labeled are two either 0 or 1 keeping the context same the quantum bit usually called as qubit is analogous to a bit having the capability to represent 3 states instead of two of that of bit. But note that like a bit quit can also have 2 state but that become obscure and ambiguous as we see the superposition phenomenon. Those two states are generally called as THE SPINS of subatomic particles and their notifications are |0> for negative spin and |1> for the positive spin one may ask a question whats the difference between bit and qubit as they are same with number of states (0 and 1) but there is one more state which is the most important called as SUPERPOSITION STATE . This state is the linear combination of those two state |0> and |1>.

And which is represented by ψ and the linear equation which represents the superposition state is

|ψ>=α |0> +β|1>.

Where α and β are the two complex numbers generalized as a+ib. where i^2=1.

While a qubit can exist in a superposition of 0 or 1 as I have seen in earlier post due to the property of uncertainty if we do the measurement we aren’t going to find the exact state of superposition because position and momentum of quantum particles cannot be measured simultaneously so we calculate it in the form of probability distribution .

|α^2>= Tells us the probability of finding the superposition | ψ> in 0^th state.

|β^2>= Tells us the probability of finding the superposition | ψ> in 1^st state.

Mixed State:

A mixed state is somewhat different than the pure one. It cannot be described by the ket or a vector instead of that it can be represented by a density matrix , and it is denoted by a symbol ρ we can say that it’s a generalize state equation because it can represent pure state as well as mixed and the density matrix is defined as

|In which Ps is the part of the two different superposition’s .to check whether the state is pure or mixed one should find the trace of p^2 if the trace is 1 then it is a pure state and if it is less than 1 then it is a mixed state . Note that the trace cannot be greater than 1 because we are dealing with probability distribution and it cannot be greater than 1.

Quantum States Part 1.

In the last post we had the basic introduction to quantum computing and the need of it ,But to develop quantum algorithms and logic one must have the all knowledge of the quantum states of sub atomic particles , because quantum computing is the integration of particle physics and computing (as said by Richard Feynman ) Before going to the actual states we need to understand the energy fundamentals associated with the subatomic particles (how they act so weird).
Before 19th century people used to believe that energy is continuous in nature but in 19th century MAX PLANK proved that energy always present in discreet packets called quanta . this is really important because if we think that energy is continuous we can cut it in piece of value that we want and the absoluteness of the electron or photon would had no significance and there wouldn’t be any need for studying photoelectric effect or Compton effect. In other words I would be saying ‘I want to buy 1.5 shirt or 2.74569 laptops ‘ .
PlANKS CONSTANT:
h =6.626*10^-34 j.s (note that energu is always multiplied by seconds in this constat)
but after some advance research it has found that whenever we use the planks constant in equation we divide it by 2π . It is all related to the angular motion which electronics possess .
so the new value of planks constant is h1=h/ 2π.
Which is 1.05456*10^-34.
Quantum States are basically the positions possessed by the subatomic particles at certain event ‘t’. As per the uncertainty principle its impossible to determine the position and momentum of the electron simultaneously , since the phenomenon is equivocal ,but we can define the state by putting them in a mathematical tool called as KET , KET is a collection of the no of states of subatomic particles, for an example we will take a hydrogen atom , in which I,j,k variables describes the state of the electron it may be a pure state or it may be mixed , but one important thing about these states are they always been calculated as the probability function because practically its almost impossible to locate or track the exact position of electron they may be present at two different places at same time , for example if ask you that whats the probability of you going Jupiter with no time ? you may call me insane for asking this stupid question but there is a valid number it may ->0 but still there is a probability . It may make you to think about parallel universe and STRING THEORY , MEMBERANE THEORY etc.
So the states are subdivided into two groups
1) Pure State.
2) Mixed States.
Pure states are extreme pure a positive spin ,negative spin or a superposition
While a mixed statistical state is mixture of two pure its complex and depends on probability function of two different pure states
As the states are need to explained by formulating them in actual KET we will discuss the later part in the next article .