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- I can't get rid of constant phase factors when superposing wave functions. What am I not getting?
I'm trying to repair the dismal state of my knowledge of QM, so I downloaded Tong's notes and have read through them a couple of times and I have a question.
Tong says (section 1.1.1, p7 in the pdf) that an overall constant phase factor ##e^{i\alpha}## infront of a wavefunction describes an equivalent state - ##\psi(x,t)\equiv e^{i\alpha}\psi(x,t)##. In particular, he notes that the observable probability density function is ##P(x,t)=\left(e^{i\alpha}\psi(x,t)\right)\left(e^{i\alpha}\psi(x,t)\right)^*##, and the complex conjugation causes the exponentials to cancel. OK, fine.
My question is about superposition. If I superpose two states ##e^{i\alpha}\psi_\alpha## and ##e^{i\beta}\psi_\beta## and compute the probability density of the superposed state I get$$\begin{eqnarray*}
P&=&\left(e^{i\alpha}\psi_\alpha+e^{i\beta}\psi_\beta\right)\left(e^{i\alpha}\psi_\alpha+e^{i\beta}\psi_\beta\right)^*\\
&=&|\psi_\alpha|^2+e^{i(\alpha-\beta)}\psi_\alpha\psi^*_\beta+e^{i(\beta-\alpha)}\psi_\beta\psi^*_\alpha+|\psi_\beta|^2\\
&=&|\psi_\alpha|^2+2\mathrm{Re}\left(e^{i(\alpha-\beta)}\psi_\alpha\psi^*_\beta\right)+|\psi_\beta|^2
\end{eqnarray*}$$which does depend on the phase factors in front of the contributing states. To me, that implies that those states aren't equivalent. Am I misunderstanding something? Reading too much into "equivalent"? Making a stupid arithmetic error?
Tong says (section 1.1.1, p7 in the pdf) that an overall constant phase factor ##e^{i\alpha}## infront of a wavefunction describes an equivalent state - ##\psi(x,t)\equiv e^{i\alpha}\psi(x,t)##. In particular, he notes that the observable probability density function is ##P(x,t)=\left(e^{i\alpha}\psi(x,t)\right)\left(e^{i\alpha}\psi(x,t)\right)^*##, and the complex conjugation causes the exponentials to cancel. OK, fine.
My question is about superposition. If I superpose two states ##e^{i\alpha}\psi_\alpha## and ##e^{i\beta}\psi_\beta## and compute the probability density of the superposed state I get$$\begin{eqnarray*}
P&=&\left(e^{i\alpha}\psi_\alpha+e^{i\beta}\psi_\beta\right)\left(e^{i\alpha}\psi_\alpha+e^{i\beta}\psi_\beta\right)^*\\
&=&|\psi_\alpha|^2+e^{i(\alpha-\beta)}\psi_\alpha\psi^*_\beta+e^{i(\beta-\alpha)}\psi_\beta\psi^*_\alpha+|\psi_\beta|^2\\
&=&|\psi_\alpha|^2+2\mathrm{Re}\left(e^{i(\alpha-\beta)}\psi_\alpha\psi^*_\beta\right)+|\psi_\beta|^2
\end{eqnarray*}$$which does depend on the phase factors in front of the contributing states. To me, that implies that those states aren't equivalent. Am I misunderstanding something? Reading too much into "equivalent"? Making a stupid arithmetic error?
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