Work done by introducing a spin in supersposition into a Magnetic Field

A spin is created in a superposition of up and down states. A magnet is moved very slowly, towards the spin. What is the work done by the magnet. It may be helpful to imagine that the magnet is connected to a spring that expands or contracts depending on work done by the magnet.

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You tagged this "measurement problem" which I don't see to match at all. Is this homework? – A.O.Tell Oct 6 '12 at 14:25
No this is not homework. I think a consideration like this have some relevance to measurement problem. Because I believe that from the sign of the work done one can infer the spin. – Prathyush Oct 6 '12 at 15:37
So find the energy, find its change, get work. And I still don't see any relation to the measurement problem – A.O.Tell Oct 6 '12 at 15:39
I think Ive been rather brief, The change in the expectation value of energy cannot be the work done. Energy can be infact Zero if the particle is in a Half and Half superposition of up and down states. The concept of work being a thermodynamic cannot be equal to change in energy. I think One must analyze this problem by considering a statistical system like a spring, which is composed of many degrees of freedom. So spring is connected to a magnet. When the Spring + Magnet system in bought in the presence of spin, this system expands or contracts, Measuring the spin of the particle. – Prathyush Oct 6 '12 at 15:56
What you propose is not in agreement with quantum theory – A.O.Tell Oct 6 '12 at 15:59

The work is of course equal to the potential energy of the dipole $\vec m$ in the magnetic field, $$U = -\vec m \cdot \vec B.$$ Now, your wording indicates that the magnetic field $\vec B$ is macroscopic so it may be treated as a classical parameter. However, $\vec m$ from a spinning particle is quantum mechanical. For example, for an electron, $$\vec m = -g_S \mu_B \frac{\vec S}{\hbar}$$ where the spin $g$-factor is $g_S\sim 2.0023$, $\mu_B=e\hbar/(2m_e)$ is the Bohr magneton, and $\vec S$ is the operator (or triplet of operators) of the electron's angular momentum – that act as $\hbar/2$ times the Pauli matrices on the spin-up and spin-down states.

So up to a multiplicative constant $\gamma$, $$U = \gamma \vec \sigma\cdot \vec B .$$ The work done on the electron is a $q$-number, an operator, which acts on the spin-up and spin-down states. For the up and down states relatively to the direction of the external magnetic field $\vec B$, the work done has a sharply defined value, an eigenvalue of $U$. For general linear superpositions, the work done on the electron has a certain probability amplitude to be positive and a certain probability amplitude to be negative. The squared absolute values of these probability amplitudes determine the probabilities that the work will be seen to be one (positive) value or the other (negative) value, just like everywhere in quantum mechanics.

It's however very interesting to consider superpositions of states in this context, especially for $j=1/2$. For spin-1/2 particles, each superposition of up and down states is equivalent to "up" with respect to a certain axis in space. If we deal with general superpositions, this axis on which the spin is "up" will generally not be aligned with the direction of the magnetic field. If that's the case, the presence of the magnetic field will have the effect of causing "precession" of the axis defining the spin up. The axis at which the particle is polarized "up" will be "turning around" the direction of $\vec B$ like a ninny. This precession results from the time-dependent change of the phases of the wave function – which is opposite for the up and down states, so the relative phase is changing as well – and the fact that the relative phases of the wave functions always matter (for something) in quantum mechanics.

The classic experiment measuring this spin-dependent magnetic work and exhibiting lots of the quantum properties is called the Stern-Gerlach experiment. With some probabilities, the particle behaves in one way or another.

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Thank you, I followed what you said. "For general linear superpositions, the work done on the electron has a certain probability amplitude to be positive and a certain probability amplitude to be negative." I was wondering if one can analyze this further by constructing some sort of statistical apparatus. The most natural one that comes to my mind is a spring attached to a magnet. Spring being composed of many degrees of freedom, one can apply the laws of quantum mechanics to each degree of freedom, But what is Classically measurable is only the extension. Motive was to understand measurement. – Prathyush Oct 6 '12 at 19:37
Apologies, I am not sure whether I understand the question. If you want to know something about measurement in quantum mechanics, that's a frequently discussed "controversial" issue but there's nothing unknown about it today. A macroscopic apparatus correlates/entangles the measured degree of freedom, in this case a spin, with some of its macroscopic properties that decohere. This diagonalizes the density matrix so the probabilities may be interpreted in the same way as in classical physics. – Luboš Motl Oct 8 '12 at 5:22
I understand what happens to the system under observation, after decoherence. The off-diagonal terms of the density matrix of the system containing superposition terms are sent to zero and the basis is chosen by the interaction with apparatus. But what I don't understand is how this information is recorded by the apparatus. The apparatus somehow in this process, records information that is accessible classically.(such as moving of a pointer). How does de-coherence imply a classically observable change in the state of the apparatus? – Prathyush Oct 8 '12 at 23:37
This is a slightly off topic, but I think has some relevance to the Issue of interest. To me it appears that there is something Thermodynamic about measurement apparatus. An example to illustrate this point is A Cold heat bath coupled to a system with 2 energy Eigenstates can measure it. The Bath has a slight change in Temperature, which can be monitored using classically probes, like thermometers. The Energy of the system is equal to Specific heat*Change in temperature. If you say that de-coherence is complete description of measurement, then perhaps I am missing some important aspect of it. – Prathyush Oct 9 '12 at 6:56