- #1

peteratcam

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Following a short discussion in another thread, I promised to start a new thread for this.

I asked myself the question "how would I describe the process of measurement in a fully quantum mechanical way?" and came up with this.

1) A model for the general measuring device

The aim is to have a QM system which describes a measuring device, with a finite dimensional space of states so that the whole thing can be written down with finite matrices.

I imagine a dial, with a pointer (see picture coming soon).

Any position of the pointer corresponds to a different quantum mechanical state for the pointer (much like a 1-d particle on a ring).

To make things easier, discretise the possible positions of the pointer, so that we have N possible pointer states, all equally spaced.

Assume that the Hamiltonian for the dial is H = 0 so there is no dynamics at all.

So the measuring apparatus is an N-state quantum system.

2) Operators for the measuring device.

There is only one operator we need for the measuring device: the operator that moves the pointer. By analogy with the momentum operator, we want the generator of pointer rotations, which I will call R.

Define it so that the unitary matrix exp(-iR) advances the pointer clockwise by pi radians.

So far so good, we have a device which we can start in the state pointing to 0, and we have an operator which rotates the pointer round by whatever angle we want.

We can calibrate the device by letting the maximum reading be 'm', corresponding to pi radians.

3) Performing a measurement of the number '4'.

Suppose we want to measure a number, called y, using our device, the way to do it is:

In units where hbar = 1, change the hamiltonian to H = R y/m for a duration of 1 unit of time, then change back to H = 0.

The time evolution operator will be exp(-i(y/m)R), which will swing the pointer round by (y/m) of a half circle.

Eg, we might choose m = 20, with N = 40, so the states on the dial run from -19,-18,....0,...19,20.

Acting on the initial state with exp(-i(4/20)R) will move the pointer to point at '4'. ie, we have measured the number 4 using our measuring apparatus.

[EDIT at 9am GMT: Despite numerous replies already, I will continue from here, but address concerns in a reply below]

4) Measuring the properties of another quantum system

So far we have shown how to measure a number on our dial. Now we want to measure the properties of another system.

Generalising what we did to measure a number, the following procedure will 'measure' the value of observable [tex]\hat O[/tex]. (I've put the hat on it so it doesn't look like zero).

The full hamiltonian, which acts on the product space [tex]{\mathcal H}_M\otimes{\mathcal H}_S[/tex] will

initially be [tex]H = \mathbf 1 \otimes H_S[/tex] since our measuring device has H=0.

Now turn on the measurement hamiltonian. Again, 'm' sets the maximum reading on the dial.

[tex]H_{meas} = \frac{1}{m\tau}R\otimes \hat O[/tex]

After time tau, set the measurement hamiltonian back to H = 0. Ideally, our clever experimentalist is able to make 'tau' very short, so we have an instantaneous measurement.

4a) Explicit example, measuring the energy of an harmonic oscillator.

Suppose the maximum reading on my dial is 3, and I have N = 12. Label the states of the apparatus by [tex]|-2.5\rangle,|-2.0\rangle,|-1.5\rangle,\ldots|0.0\rangle,\ldots,|2.5\rangle,|3.0\rangle [/tex].

Also, suppose the system we are interested in is a Harmonic oscillator, currently in the state

[tex]\frac{1}{2}|0\rangle+\frac{1}{\sqrt{2}}|1\rangle+\frac{1}{2}|2\rangle[/tex].

Therefore the inital state of the composite system is the product state:

[tex]|0.0\rangle \otimes [\frac{1}{2}|0\rangle+\frac{1}{\sqrt{2}}|1\rangle+\frac{1}{2}|2\rangle][/tex]

Now the experimentalist comes along, and attaches his measuring device to the harmonic oscillator in such a way as to measure the energy. The effect of this is that the measurement hamiltonian becomes

[tex]H_{meas} = \frac{1}{m\tau}R\otimes H_s[/tex].

After time tau (which is much shorter than the period of the oscillator), he disconnects the measuring device.

The final state of the composite system is

[tex]\frac{1}{2}|0.5\rangle \otimes |0\rangle+\frac{1}{\sqrt{2}}|1.5\rangle \otimes |1\rangle+\frac{1}{2}|2.5\rangle \otimes |2\rangle[/tex]

The apparatus state is now entangled with the measured state.

*So the readings on the apparatus are the eigenvalues of the observable.

*Moreover, the state of the system is the eigenvector for the measured value.

Crucially, the apparatus states are orthogonal, so there is no interference possible between the two options for the system.

Applying the probability interpretation to the entangled system is consistent with applying it to just the system we measured originally, except the measuring process and apparatus are described in a QM way too now.

I believe the points with (*) were the points in the other thread under point 2, which I wanted to address.

5) Generalisations.

This is completely general. My measuring apparatus will measure any observable you like and in performing the measurement, it is clear why only eigenvalues are allowed, and why eigenvectors is what you get afterwards.

[edit: more may need to be edited if it is unclear, but I thought it better to have all of it in one place if there is going to be a discussion]

I asked myself the question "how would I describe the process of measurement in a fully quantum mechanical way?" and came up with this.

1) A model for the general measuring device

The aim is to have a QM system which describes a measuring device, with a finite dimensional space of states so that the whole thing can be written down with finite matrices.

I imagine a dial, with a pointer (see picture coming soon).

Any position of the pointer corresponds to a different quantum mechanical state for the pointer (much like a 1-d particle on a ring).

To make things easier, discretise the possible positions of the pointer, so that we have N possible pointer states, all equally spaced.

Assume that the Hamiltonian for the dial is H = 0 so there is no dynamics at all.

So the measuring apparatus is an N-state quantum system.

2) Operators for the measuring device.

There is only one operator we need for the measuring device: the operator that moves the pointer. By analogy with the momentum operator, we want the generator of pointer rotations, which I will call R.

Define it so that the unitary matrix exp(-iR) advances the pointer clockwise by pi radians.

So far so good, we have a device which we can start in the state pointing to 0, and we have an operator which rotates the pointer round by whatever angle we want.

We can calibrate the device by letting the maximum reading be 'm', corresponding to pi radians.

3) Performing a measurement of the number '4'.

Suppose we want to measure a number, called y, using our device, the way to do it is:

In units where hbar = 1, change the hamiltonian to H = R y/m for a duration of 1 unit of time, then change back to H = 0.

The time evolution operator will be exp(-i(y/m)R), which will swing the pointer round by (y/m) of a half circle.

Eg, we might choose m = 20, with N = 40, so the states on the dial run from -19,-18,....0,...19,20.

Acting on the initial state with exp(-i(4/20)R) will move the pointer to point at '4'. ie, we have measured the number 4 using our measuring apparatus.

[EDIT at 9am GMT: Despite numerous replies already, I will continue from here, but address concerns in a reply below]

4) Measuring the properties of another quantum system

So far we have shown how to measure a number on our dial. Now we want to measure the properties of another system.

Generalising what we did to measure a number, the following procedure will 'measure' the value of observable [tex]\hat O[/tex]. (I've put the hat on it so it doesn't look like zero).

The full hamiltonian, which acts on the product space [tex]{\mathcal H}_M\otimes{\mathcal H}_S[/tex] will

initially be [tex]H = \mathbf 1 \otimes H_S[/tex] since our measuring device has H=0.

Now turn on the measurement hamiltonian. Again, 'm' sets the maximum reading on the dial.

[tex]H_{meas} = \frac{1}{m\tau}R\otimes \hat O[/tex]

After time tau, set the measurement hamiltonian back to H = 0. Ideally, our clever experimentalist is able to make 'tau' very short, so we have an instantaneous measurement.

4a) Explicit example, measuring the energy of an harmonic oscillator.

Suppose the maximum reading on my dial is 3, and I have N = 12. Label the states of the apparatus by [tex]|-2.5\rangle,|-2.0\rangle,|-1.5\rangle,\ldots|0.0\rangle,\ldots,|2.5\rangle,|3.0\rangle [/tex].

Also, suppose the system we are interested in is a Harmonic oscillator, currently in the state

[tex]\frac{1}{2}|0\rangle+\frac{1}{\sqrt{2}}|1\rangle+\frac{1}{2}|2\rangle[/tex].

Therefore the inital state of the composite system is the product state:

[tex]|0.0\rangle \otimes [\frac{1}{2}|0\rangle+\frac{1}{\sqrt{2}}|1\rangle+\frac{1}{2}|2\rangle][/tex]

Now the experimentalist comes along, and attaches his measuring device to the harmonic oscillator in such a way as to measure the energy. The effect of this is that the measurement hamiltonian becomes

[tex]H_{meas} = \frac{1}{m\tau}R\otimes H_s[/tex].

After time tau (which is much shorter than the period of the oscillator), he disconnects the measuring device.

The final state of the composite system is

[tex]\frac{1}{2}|0.5\rangle \otimes |0\rangle+\frac{1}{\sqrt{2}}|1.5\rangle \otimes |1\rangle+\frac{1}{2}|2.5\rangle \otimes |2\rangle[/tex]

The apparatus state is now entangled with the measured state.

*So the readings on the apparatus are the eigenvalues of the observable.

*Moreover, the state of the system is the eigenvector for the measured value.

Crucially, the apparatus states are orthogonal, so there is no interference possible between the two options for the system.

Applying the probability interpretation to the entangled system is consistent with applying it to just the system we measured originally, except the measuring process and apparatus are described in a QM way too now.

I believe the points with (*) were the points in the other thread under point 2, which I wanted to address.

5) Generalisations.

This is completely general. My measuring apparatus will measure any observable you like and in performing the measurement, it is clear why only eigenvalues are allowed, and why eigenvectors is what you get afterwards.

[edit: more may need to be edited if it is unclear, but I thought it better to have all of it in one place if there is going to be a discussion]

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