Quantum – Theory – State Equations

This post covers the generic equations for quantum states and gives the equations for the poles on each of the axes. As the Wave Number Bloch sphere is not available in our shop, references to the Bloch sphere will include both the Wave Number and Bloch sphere rotation ball co-ordinates where used.

Standard Symbols

Ψ, ψ, psi is the quantum state of a qubit.

Φ, ϕ, phi, is the angle measured counterclockwise from the x axis.

𝜭, theta, is the angle down from the z axis.

i^{^} is a counterclockwise rotation of 90^{^o} around the z-axis and i^v is a clockwise rotation.

Generic Formula for Quantum States

|ψ> = cos (𝜭/2) |jˆ>  + e^iϕ sin( 𝜭/2) |j^v> is the generic formula for a quantum state.

𝜭 is the angle swept down from Z⁺/Z^{^} on the Bloch Sphere. There is no sweep from the location Z⁺/Z^{^}  to itself at Z⁺/Z^{^}, so 𝜭 = 0. A 180^{^o} angle sweeps down between Z⁺/Z^{^} and Z^–/Z^v as it is at the opposite side of the prop, so 𝜭 =180^{^o}.

A 90^{^o} angle sweeps down between  Z⁺/Z^{^} and X⁺/X^{^}, X^–/X^v, Y⁺/Y^{^} and Y^–/Y^v. Note that these 4 locations are directly below Z⁺/Z^{^} on the rotation ball.

Φ is the angle measured counterclockwise from the x-axis. The counterclockwise refers to the counterclockwise direction on the z-axis. The Y⁺/Y^{^} point is 90^{^o} counterclockwise from/X^{^}.

Continue on counterclockwise and X^–/X^v is found 180^{^o} counterclockwise and Y^–/Y^v is found 270^{^o} counterclockwise from X⁺/X^{^}.  Although Z⁺/Z^{^} and Z^–/Z^v are above and below X⁺/X^{^} on the rotation ball, the angle Φ is 0.

Quantum State Equations for Axes

The states |jˆ> and |j^v>form the basis used for these state equations.

z-axis

• |North Pole>: 𝜭 = 0; ϕ = 0
  • = cos(0/2)|jˆ> + eⁱ⁰sin(0/2)|j^v>
    • = 1|j^{^}> + 0|j^v>
      • Note eⁱ⁰ = cos(0) + isin(0) = 1  + 0 = 1
      • sin(0) = 0, so eⁱ⁰sin(0/2) = 0
    • = 1|j^{^}> =   |j^{^}>

• |South Pole>: 𝜭 = π^{^}; ϕ = 0
  • = cos(π^{^}/2)|j^{^}> + eⁱ⁰sin(π^{^}/2)|j^v>
    • = 0*|j^{^}> +  1*1*|j^v>  
    • =    |j^v> 

x-axis

• |Near Pole>: 𝜭 = π^{^}/2; ϕ = 0
  • = cos(π^{^}/4)|j^{^}> + eⁱ⁰sin(π^{^}/4)|j^v>  
    • = 1/√2|jˆ> + 1/√2|j^v>
    • = 1(|jˆ> + |j^v>)/√2
    • = (|jˆ> + |j^v>)/√2

• |Far Pole>: 𝜭 = π^{^}/2; ϕ = π^{^}  
  • = cos(π^{^}/4)|jˆ> + e^{iπ^}sin(π^{^}/4)|j^v>
    • = 1/√2|jˆ> + ^–1/√2|j^v>
      • Note e^{iπ^} = cos(π^{^}) + isin(π^{^}) = ^–1 + 0 =^–1
    • =  1(|jˆ> + ^–1|j^v>)/√2
    • = (|jˆ> + ^–|j^v>)/√2

y-axis

• |East Pole>: 𝜭 = π^{^}/2; ϕ = π^{^}/2
  • = cos(π^{^}/4)|jˆ> + e^{iπ^/2}sin(π^{^}/4)|j^v>
    • = 1/√2|jˆ> + isin(π^{^}/4))|j^v>       
      • Note e^{iπ^/2} = cos(π^{^}/2) + isin(π^{^}/2) = 0 + i*1^{^} = i^{^}
    • = 1/√2|jˆ> +   i/√2)|j^v>
    • = (|jˆ> + i|j^v>)/√2

• |West Pole>: 𝜭 = π^{^}/2; ϕ = 3π^{^}/2 
  • = cos(π^{^}/4)|jˆ> + e^{i3π^/2}sin(π^{^}/4)|jv>
    •  = 1/√2|jˆ> + isin(3π^{^}/4)|j^v>  
      • Note e^{i3π^/2} = cos(3π^{^}/2) + isin(3π^{^}/2) = 0 + i*^–1 = ^–i
    • = 1/√2|jˆ> +  ^–i/√2|j^v>
    • = (|jˆ> + ^–i|j^v>)/√2 

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