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1. B How does quantum tunneling occur without an observer?

Tunnelling indeed involves measurement. If we try to find the position of the electron a large number of times, most of the times it will be found inside the barrier meaning that the state of the quantum mechanical system has collapsed to a state corresponding to the position inside the...
2. Microcanonical ensemble density matrix

Perhaps this whole thing is true for a discrete basis...
3. Microcanonical ensemble density matrix

Let us see what goes wrong in the following reasoning. 1) 2) So in the energy representation, the microcanonical density matrix has the following form. \rho_{m,n} = \frac{1}{N} \delta_{m,n} . Clearly here it is proportional to the identity operator. Now the representation in any...
4. Microcanonical ensemble density matrix

Not the Canonical density matrix. I was referring solely to microcanonical ensemble.
5. Microcanonical ensemble density matrix

Here is another explanation from a correspondence with one of the authors. Now this seems simple. If the density matrix is proportional to the identity matrix in the energy representation then it has to be so in any other representation. This is because the matrices in different bases have...
6. Microcanonical ensemble density matrix

In a correspondence with one of my professors I understand that observables cannot distinguish states within the microcanonical subspace. So the density matrix in this subspace is always going to be proportional to the identity matrix. Also I see that what vanhees71 says here has to be...
7. Microcanonical ensemble density matrix

Yes. It is now that Pathria invokes the postulate of random phases to make it diagonal. The average of c_n c_m ^* becomes zero due to random phases among of the states. ie. \langle c_n c_m^* \rangle = c_nc_m^* \langle e^{i(\theta_n - \theta_m)} \rangle = c_n c_m^* \delta_{nm}= |c_n|^2. ...
8. Microcanonical ensemble density matrix

That's right. But The density matrix is made diagonal in this book by invoking the postulate of random phases. The diagonal elements are said to be equal due to the postulate of equal a priori probabilities. My question then is this. Does the postulate of equal a priori probabilities imply...
9. Microcanonical ensemble density matrix

What I find most confusing is the equality of the diagonal elements of density matrix in all representations. I am not able to understand this.
10. Microcanonical ensemble density matrix

Yes, I found the postulate of random phases being applied to energy representation else where too. But Pathria has the density matrix automatically diagonal in energy basis and makes it diagonal in all other basis by applying this postulate. This is what I find confusing. In lecture 9 of the...
11. Microcanonical ensemble density matrix

Thanks. But Pathria does say it. Please refer to page 119,120 in statistical mechanics third edition by Pathria. These pages correspond to section 5.2. It is given so in other editions too. (page 118 in first edition, page 108 in second edition) In 119, \rho_{mn} = \rho_n \delta_{mn} -------...
12. Microcanonical ensemble density matrix

Ref: R.K Pathria Statistical mechanics (third edition sec 5.2A) First it is argued that the density matrix for microcanonical will be diagonal with all diagonal elements equal in the energy representation. Then it is said that this general form should remain the same in all representations. i.e...