SCEST21: Schrodinger's Cat, and Einstein's Space-time, in the 21st Century
A blogspot for discussing the connection between quantum foundations and quantum gravity
Managed by: Tejinder Pal Singh, Physicist, Tata Institute of Fundamental Research, Mumbai
If you are a professional researcher / student researching on these topics, and would like to post an article here with you as author, you are welcome to do so. Please e-mail your write-up to tpsingh@tifr.res.in and it will be uploaded here.
Keywords: Quantum foundations; Quantum gravity; Schrodinger's cat; Spontaneous collapse theory
________________________________________________
November 23, 2019
The quantum measurement problem
Tejinder Singh
A non-relativistic quantum system evolves according to the Schrodinger equation, which is linear and deterministic. Given the initial state, the final state can be precisely determined, by solving the Schrodinger equation. And if the initial state is a superposition of two eigenstates of some observable, it will evolve into a final state which is also a superposition of those two eigenstates. Moreover, if the equation is being considered for an object of some mass, the value of the mass can be arbitrarily large, according to the Schrodinger equation.
Keeping all this in mind, let us consider what happens when a quantum system, being described by the Schrodinger equation, meets a measuring apparatus. Say the apparatus measures spin of an incoming quantum particle, which is in a linear superposition of two states, say |spin up >, and
|spin down >, having a complex amplitude A to be in up state, and amplitude B to be in down state. Suppose the spin value is deduced from the position of a pointer, and the pointer points up if the state is spin-up, and the pointer points down if the state is spin-down. According to the Schrodinger equation, if the particle is in a superposition, the following state should be observed after the measurement has been done:
A |spin up> |pointer up > + B |spin down > |pointer down >
[Such a state is called an entangled state. The quantum particle and the apparatus have become `entangled’ after interacting]. The superposed state of the particle should force the pointer also into a superposed state. However, what is actually seen after the measurement is something completely different, and extremely surprising.
After the measurement the quantum particle is found either in up state with pointer pointing up, or in down state with pointer pointing down. Which of the two? Its random. The outcome is random, and cannot be predicted beforehand. If one does the same experiment many many times, sometimes the outcome is up, sometimes it is down. But the fraction of times the outcome is up, is experimentally found to be given by |A|^2, the square modulus of A. The fraction of times the outcome is down, is given by |B|^2. This is the so-called Born probability rule - an empirical rule always found to hold in quantum measurements, namely that the probability of an outcome is given by the square modulus of the corresponding amplitude.
As you can see, what actually happens during a measurement completely disagrees with the Schrodinger equation. Quantum mechanics fails during the measurement process. It fails on the following counts: (i) Superposition is lost, whereas it should not have been lost. Why did `up + down’ go to either up or down, even though the equation Is linear? (ii) The Schrodinger equation is deterministic. Why then are the outcomes random and unpredictable? (iii) Since the equation is deterministic, it has nothing to do with probabilities! Where have the probabilities arisen from? Why does the Born rule hold - it is an experimentally observed rule, which obviously cannot be derived from the Schrodinger equation. And why do the probabilities mysteriously depend on the amplitudes A and B, when probability has nothing to do with Schrodinger equation, whereas A and B are properties of the state that evolves according to the Schrodinger equation? This set of disagreements between theory and experiment is commonly referred to as the quantum measurement problem.
As we said at the beginning of the first post, if a theory agrees with some data, but not with all data, it should be replaced by a new theory which agrees with all of the data. The Schrodinger equation correctly describes the experimentally observed motion of the particle before it meets the measuring apparatus, but fails to describe what happened during the measurement process. Hence, it should be replaced by a new equation which agrees with the Schrodinger equation before the measurement (i.e. when only the quantum particle is being considered). But the new equation should disagree with the Schrodinger equation during the measurement process, in such a way that the new equation resolves the three counts on which the Schrodinger equation fails, and explains what actually happens during a measurement.
The GRWP theory of spontaneous collapse provides precisely the correct new equation for this purpose. As we know, for the quantum particle, GRWP agrees very well with quantum mechanics and with the Schrodinger equation, because the superposition life-time (being of the order T) is so large as to be practically infinite. However, now let us apply GRWP to the measurement process, and in particular to the entangled state written above. This is a superposition of two position states of the pointer (up and down). But the pointer is made of enormously many particles, and we know according to GRWP that the superposed state for such a large object is extremely short-lived, lasting only for a millionth of a second or so. After this much time the pointer spontaneously and randomly collapses to the up state or to the down state, and takes the spinning quantum particle with it. So we infer that the spin state of the particle has randomly changed to spin-up or spin-down. This is how GRWP theory solves the quantum measurement problem.
But what about the Born probability rule? Does the GRWP theory prove this modulus-square rule? Alas, it does not. It takes the rule as an assumption, a given property of spontaneous collapse. In subsequent posts, we will see how the Born rule arises as a consequence of spontaneous quantum gravity. Note that from the point of view of the GRWP theory, there is nothing special about a quantum measurement. It is just a particular case of macroscopic position superposition, which according to GRWP is short-lived. The measurement problem is the same as the problem of macroscopic superpositions not being observed in nature, so the proposed solution to the two problems is also the same, I.e. the falsifiable GRWP theory.
A blogspot for discussing the connection between quantum foundations and quantum gravity
Managed by: Tejinder Pal Singh, Physicist, Tata Institute of Fundamental Research, Mumbai
If you are a professional researcher / student researching on these topics, and would like to post an article here with you as author, you are welcome to do so. Please e-mail your write-up to tpsingh@tifr.res.in and it will be uploaded here.
Keywords: Quantum foundations; Quantum gravity; Schrodinger's cat; Spontaneous collapse theory
________________________________________________
November 23, 2019
The quantum measurement problem
Tejinder Singh
A non-relativistic quantum system evolves according to the Schrodinger equation, which is linear and deterministic. Given the initial state, the final state can be precisely determined, by solving the Schrodinger equation. And if the initial state is a superposition of two eigenstates of some observable, it will evolve into a final state which is also a superposition of those two eigenstates. Moreover, if the equation is being considered for an object of some mass, the value of the mass can be arbitrarily large, according to the Schrodinger equation.
Keeping all this in mind, let us consider what happens when a quantum system, being described by the Schrodinger equation, meets a measuring apparatus. Say the apparatus measures spin of an incoming quantum particle, which is in a linear superposition of two states, say |spin up >, and
|spin down >, having a complex amplitude A to be in up state, and amplitude B to be in down state. Suppose the spin value is deduced from the position of a pointer, and the pointer points up if the state is spin-up, and the pointer points down if the state is spin-down. According to the Schrodinger equation, if the particle is in a superposition, the following state should be observed after the measurement has been done:
A |spin up> |pointer up > + B |spin down > |pointer down >
[Such a state is called an entangled state. The quantum particle and the apparatus have become `entangled’ after interacting]. The superposed state of the particle should force the pointer also into a superposed state. However, what is actually seen after the measurement is something completely different, and extremely surprising.
After the measurement the quantum particle is found either in up state with pointer pointing up, or in down state with pointer pointing down. Which of the two? Its random. The outcome is random, and cannot be predicted beforehand. If one does the same experiment many many times, sometimes the outcome is up, sometimes it is down. But the fraction of times the outcome is up, is experimentally found to be given by |A|^2, the square modulus of A. The fraction of times the outcome is down, is given by |B|^2. This is the so-called Born probability rule - an empirical rule always found to hold in quantum measurements, namely that the probability of an outcome is given by the square modulus of the corresponding amplitude.
As you can see, what actually happens during a measurement completely disagrees with the Schrodinger equation. Quantum mechanics fails during the measurement process. It fails on the following counts: (i) Superposition is lost, whereas it should not have been lost. Why did `up + down’ go to either up or down, even though the equation Is linear? (ii) The Schrodinger equation is deterministic. Why then are the outcomes random and unpredictable? (iii) Since the equation is deterministic, it has nothing to do with probabilities! Where have the probabilities arisen from? Why does the Born rule hold - it is an experimentally observed rule, which obviously cannot be derived from the Schrodinger equation. And why do the probabilities mysteriously depend on the amplitudes A and B, when probability has nothing to do with Schrodinger equation, whereas A and B are properties of the state that evolves according to the Schrodinger equation? This set of disagreements between theory and experiment is commonly referred to as the quantum measurement problem.
As we said at the beginning of the first post, if a theory agrees with some data, but not with all data, it should be replaced by a new theory which agrees with all of the data. The Schrodinger equation correctly describes the experimentally observed motion of the particle before it meets the measuring apparatus, but fails to describe what happened during the measurement process. Hence, it should be replaced by a new equation which agrees with the Schrodinger equation before the measurement (i.e. when only the quantum particle is being considered). But the new equation should disagree with the Schrodinger equation during the measurement process, in such a way that the new equation resolves the three counts on which the Schrodinger equation fails, and explains what actually happens during a measurement.
The GRWP theory of spontaneous collapse provides precisely the correct new equation for this purpose. As we know, for the quantum particle, GRWP agrees very well with quantum mechanics and with the Schrodinger equation, because the superposition life-time (being of the order T) is so large as to be practically infinite. However, now let us apply GRWP to the measurement process, and in particular to the entangled state written above. This is a superposition of two position states of the pointer (up and down). But the pointer is made of enormously many particles, and we know according to GRWP that the superposed state for such a large object is extremely short-lived, lasting only for a millionth of a second or so. After this much time the pointer spontaneously and randomly collapses to the up state or to the down state, and takes the spinning quantum particle with it. So we infer that the spin state of the particle has randomly changed to spin-up or spin-down. This is how GRWP theory solves the quantum measurement problem.
But what about the Born probability rule? Does the GRWP theory prove this modulus-square rule? Alas, it does not. It takes the rule as an assumption, a given property of spontaneous collapse. In subsequent posts, we will see how the Born rule arises as a consequence of spontaneous quantum gravity. Note that from the point of view of the GRWP theory, there is nothing special about a quantum measurement. It is just a particular case of macroscopic position superposition, which according to GRWP is short-lived. The measurement problem is the same as the problem of macroscopic superpositions not being observed in nature, so the proposed solution to the two problems is also the same, I.e. the falsifiable GRWP theory.
No comments:
Post a Comment
The purpose of this blog is to have a discussion on the connection between quantum foundations and quantum gravity. Students and professionals working on or interested in these subjects are very welcome to participate. Please post only on this or related topics. Off-topic comments will be removed. Obscene, vulgar and abusive posts will be removed.