# Origin of probabilities in Quantum Mechanics?

The non-normalized wavefunction of a general qubit is given by: $$|\psi\rangle=A|0\rangle+B|1\rangle.$$ The complex amplitudes $$A$$ and $$B$$ can be represented by two arrows in the complex plane:

Now the wavefunction can be multiplied by any complex number $$R$$ without changing the physics. This will cause the arrows $$A$$ and $$B$$ to rotate and shrink/expand together with a fixed angle between them.

Therefore two sets of points will be traced out represented by a circle with area $$|A|^2$$ and a circle with area $$|B|^2$$. These represent the sets of possible values for the amplitudes $$A$$ and $$B$$.

Thus if we become entangled with the qubit then the probabilities of finding ourselves in set $$A$$ (measuring $$0$$) or set $$B$$ (measuring $$1$$) are given by: $$P(0)=\frac{|A|^2}{|A|^2+|B|^2}$$ $$P(1)=\frac{|B|^2}{|A|^2+|B|^2}.$$

Does this picture help to understand the origin of probabilities in quantum mechanics?

Correction

Let $$A=R_Ae^{i\theta_A}$$ $$B=R_Be^{i\theta_B}$$ A general normalized wavefunction is given by: $$|\psi\rangle=\frac{1}{(R_A^2+R_B^2)^{1/2}}\large[R_Ae^{i\theta_A}+R_Be^{i\theta_B}\large]$$ Assume that I multiply the amplitudes $$A$$ and $$B$$ by $$C=Re^{i\theta}$$ Then the normalized wavefunction becomes $$|\psi\rangle=\frac{1}{R(R_A^2+R_B^2)^{1/2}}\large[RR_Ae^{i(\theta_A+\theta)}+RR_Be^{i(\theta_B+\theta)}\large]$$ $$|\psi\rangle=\frac{e^{i\theta}}{(R_A^2+R_B^2)^{1/2}}\large[R_Ae^{i\theta_A}+R_Be^{i\theta_B}\large]$$ It seems that the only degree of freedom is a phase angle $$\theta$$ rather than an area as I asserted above.

• Why should we associate the area of the circles you have drawn with the probabilities to measure the qubit to have value $0$ or $1$? Commented Feb 26, 2021 at 16:29
• Each circle $A$ or $B$ represents the sets of values for the amplitudes $A$ or $B$. Commented Feb 26, 2021 at 16:44
• Yes but why should the areas of these sets be related to the probability to measure a qubits to have value 0 or 1? Commented Feb 26, 2021 at 16:45
• Also how would you generalize this argument to a system of 2 entangled qubits, where the states live in 4 dimensions? Your logic would lead you to consider volumes of 4 dimensional spheres, in which case the probabilities would scale with the 4-th power of the amplitude. Commented Feb 26, 2021 at 16:52
• Actually you're right there's just an angle that can be chosen not an area. Commented Mar 3, 2021 at 17:05