Need an explanation for a quantum model [closed]

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When I am reading chapter 10 of Dasgupta I faced a paragraph that I can't understand:

An electron can be in a ground state or in an excited state. In the Dirac notation used in quantum physics, these are denoted 0 and 1. But the superposition principle says that, in fact, the electron is in a state that is a linear combination of these two: a0|0> + a1|1>. This would make immediate sense if the a's were probabilities, nonnegative real numbers adding to 1. But the superposition principle insists that they can be arbitrary complex numbers, as long as the squares of their norms add up to 1!

Can someone describe me last 3 lines?

Answer 1

I think the author is trying to point out a difference between the quantum model and the standard assumptions you might have about probabilities.

Suppose, for example, that the electron is either up or down. In a deterministic universe, it would either be 100% up or 100% down. If we assumed that the electron probabilistically chose to be either up or down with some probabilities, then we might say that, for example, the electron was 50% up and 50% down.

When using bra-ket notation for the above, you might be tempted to say that we would say that the particle is 90% up by writing

0.5 |up> + 0.5|down>

The intuition being that the electron is simultaneously 50% up and 50% down. However, this would be incorrect. When dealing with quantum states, the configuration of a particle is related to something called the wavefunction, and it is the square of the wavefunction that determines probabilities, not the wavefunction itself. Thus if we wanted to write out a quantum state where a particle has a 50% chance of being up and a 50% chance of being down, we'd represent that as

0.707 |up> + 0.707 |down>

Since 0.707 is about the square root of 0.5, so if we square the coefficients assigned to up and down we get back the classical probabilities. As long as the squares of the coefficients sum to one, the coefficients are legal since their square gives back a probability distribution.

Of course, it's actually a bit trickier than this. The coefficients in quantum states can be complex numbers as well. For example, this is a perfectly legal quantum configuration:

(0.707 + 0.707i) |up> + 0 |down>

Here, the coefficient for up is a complex number. To get the probability of seeing up, we compute the complex conjugate of the coefficient:

(0.707 + 0.707i)(0.707 - 0.707i) = (0.5 + 0.5) = 1

So in this case the probability of seeing up is 1 and the probability of seeing down is 0^2 = 0. Since this sums to one, this is a valid quantum state.

To summarize: a probability distribution is a way of assigning real-valued weights to outcomes so that the weights sum to one. A quantum state is a way of assigning complex-valued weights to outcomes so that the sum of the product of each coefficient with it's complex conjugate is one.

Phew! Haven't had to think about that in a while! Hope this helps!