- #1

- 753

- 138

Let

**H**be a two dimensional complex Hilbert space of quantum states. An

*observable*is a self adjoint operator

**A**on

**H**. Let ##(|\phi_1\rangle,|\phi_2\rangle)## be an orthonormal eigenbasis to

**A**, with corresponding (real) eigenvalues ##a_1## and ##a_2##.

Now, let us make a measurement of

**A**upon a state ##|\psi\rangle=b_1|\phi_1\rangle+b_2|\phi_2\rangle##, where ##|b_1|^2+|b_2|^2=1##.

If ##a_1\neq a_2##, then, with probability ##b_i##, the result of the measurement is ##a_i## and the state ##|\psi\rangle## is "collapsed" into the state ##|\phi_i\rangle## (##i=1,\,2##).

But if ##a_1=a_2##, then the state does

*not*collapse, but remains unchanged after the measurement: ##|\psi\rangle=b_1|\phi_1\rangle+b_2|\phi_2\rangle##.

This is how it must be, if the collapse in both cases means a normalized orthogonal projection onto an eigenspace - the difference is that we have

*two*eigenspaces if ##a_1\neq a_2##, but only

*one*if ##a_1=a_2##.

The above is my understanding after reading p. 2-3 here: http://www.theory.caltech.edu/people/preskill/ph229/notes/chap2.pdf

But if this is correct, there is a strange discontinuity in the dependence of the eigenvalues. Assume that ##b_1=b_2=1/\sqrt 2##, and ##a_1=a_2=1##. Then the state after the measurement is the same as before the measurement: ##|\psi\rangle=(1/\sqrt 2) \phi_1+(1/\sqrt 2)\phi_2##.

But if we change the eigenvalues very slightly so that ##a_1=0.999## and ##a_2=1.001##, then, after the measurement, the new state will be ##\phi_i## with probability ##1/2## ##(i=1,\,2)##.

This is not at all close to the state in the previous case, despite the very small differences in the eigenvalues.

Is there really such a discontunity, and if so, why? Or did I misunderstand something?