- #1
Toby_phys
- 26
- 0
An electron can tunnel between potential wells. Its state can be written as:
$$
|\psi\rangle=\sum^\infty_{-\infty}a_n|n\rangle
$$
Where $|n \rangle$ is the state at which it is in the $n$th potential well, n increases from left to right.
$$
a_n=\frac{1}{\sqrt{2}}\left(\frac{-i}{3}\right)^{|n|/2}e^{in\pi}
$$
What is the probability of finding the election in well $0$ or above?
Is this probability
$$|a_0|^2+|a_1|^2+...$$
or is it $$|a_0+a_2+...|^2$$?
I am leaning towards the first option but this doesn't use the exponential phase factor. My reasoning is, let $$|\phi \rangle$$ be the superposition of everything 0 and above:
$$
|\phi\rangle=\frac{\sum^{\infty}_{0}a_n|n\rangle}{\sqrt{\sum^{\infty}_0|a_n|^2}}
$$
The denominator normalises everything.
Taking the inner product we get:
$$
\langle\phi|\psi \rangle=\frac{\sum^{\infty}_{0}|a_n|^2}{\sqrt{\sum^{\infty}_0|a_n|^2}}=\sqrt{\sum^{\infty}_0|a_n|^2}
$$
The mod square of this is the sum of the individual probabilities.
$$
|\psi\rangle=\sum^\infty_{-\infty}a_n|n\rangle
$$
Where $|n \rangle$ is the state at which it is in the $n$th potential well, n increases from left to right.
$$
a_n=\frac{1}{\sqrt{2}}\left(\frac{-i}{3}\right)^{|n|/2}e^{in\pi}
$$
What is the probability of finding the election in well $0$ or above?
Is this probability
$$|a_0|^2+|a_1|^2+...$$
or is it $$|a_0+a_2+...|^2$$?
I am leaning towards the first option but this doesn't use the exponential phase factor. My reasoning is, let $$|\phi \rangle$$ be the superposition of everything 0 and above:
$$
|\phi\rangle=\frac{\sum^{\infty}_{0}a_n|n\rangle}{\sqrt{\sum^{\infty}_0|a_n|^2}}
$$
The denominator normalises everything.
Taking the inner product we get:
$$
\langle\phi|\psi \rangle=\frac{\sum^{\infty}_{0}|a_n|^2}{\sqrt{\sum^{\infty}_0|a_n|^2}}=\sqrt{\sum^{\infty}_0|a_n|^2}
$$
The mod square of this is the sum of the individual probabilities.