A New Interpretation of Quantum Mechanics

The standard Copenhagen Interpretation of Quantum Mechanics is under challenge mainly as a result of recent experiments on phenomena leading to quantum computations. It is now known that systems could exist in superposition states that have been used in the incipient field of quantum computing. We give below the postulates of a new interpretation.

1.      A state of a quantum mechanical system is represented by a vector (ray) c in Hilbert space, where c can be expressed as linear combinations of eigenvectors in Hilbert space, of Hermitian operators corresponding to observables. A state of a quantum mechanical system can be represented by different linear combinations corresponding to different modes. Thus a state could have a number of modes, each observable (Hermitian operator) corresponding to one mode.

2.      If c is expressed as a linear combination of two or more of the eigenvectors of an Hermitian operator then the corresponding observable cannot be observed (or measured) by a human observer with or without the aid of an apparatus. In other words the mode of the particular observable cannot be observed and a value cannot be given to the observable, which also means that no measurement is made on the observable. 

3.      However, the non observation of a mode does not mean that the mode does not “exist”. We make a distinction between the “existence” of a mode, and the observation of a mode with or without the aid of an apparatus. A mode in respect of an Hermitian operator could “exist” without being observed. The knowledge of the “existence” of a mode is independent of its observation or measurement. In other words the knowledge of the “existence” of a mode of a quantum mechanical state is different from the knowledge of the value that the observable corresponding to the relevant Hermitian operator would take.

4.      If a mode of a quantum mechanical state is represented by an eigenvector, and not as a linear combination of two or more eigenvectors, of an Hermitian operator, then the mode could be observed by a human observer with or without the aid of an apparatus, and the value of the corresponding observable (or the measured value) is given by the eigenvalue which the eigenvector belongs to. It has to be emphasised that only those modes of a quantum mechanical state that can each be represented by an eigenvector, and not by a linear combination of eigenvectors, of an Hermitian operator can be observed.

5.      If a mode of a quantum mechanical state is represented by an eigenvector of an Hermitian operator then the mode corresponding to the conjugate operator cannot be represented by an eigenvector of the conjugate operator. It can be expressed as a linear combination of two or more of the eigenvectors of the conjugate operator. This means that the observable corresponding to the conjugate operator cannot be observed, or in other words it cannot be measured. However, the relevant mode corresponding to the conjugate operator can be represented as a linear combination of two or more eigenvectors of the conjugate operator, and the mode “exists” though it cannot be observed.   

6.      It is not necessary that at least one of the modes corresponding to two conjugate   operators should be represented by an eigenvector of the relevant operator. It is possible that both modes are represented by linear combinations of two or more eigenvectors of the respective operators.

7.      A state of a quantum mechanical system can be altered by making an operation that changes a mode or modes of the state. However, not all operations correspond to measurements or observations. Only those operations that would result in a mode being expressed as an eigenvalue and not as a linear combination of the eigenvalues, of an operator would result in measurements. 

8.      A particle entangled with one or more other particles is in general represented by a linear combination of eigenvectors of an Hermitian operator with respect to a mode. If the number of entangled particles is less than the dimension of the space of the eigenvectors of the Hermitian operator, then if a measurement is made in the particular mode, the particle would be represented by one of the eigenvectors, while the other particles entangled with it would be each represented by a different eigenvector of the Hermitian operator. However, if the number of entabled particles is greater than the dimension of the space of the eigenvectors, then in some cases, more than one particle would be represented by an eigenvector.

Thus a quantum mechanical state can exist in different modes. Certain modes may be observable while others are not, though they “exist”. It may be that there are systems for which no mode is observable.  If a mode is observable then the mode corresponding to the conjugate operator (conjugate mode) “exists” but is not observable. The position and the momentum of a particle constitute the best known example of two conjugate modes.

Suraj Chandana.

Department of Physics, University of Kelaniya, Kelaniya, Sri Lanka.

and

Nalin de Silva.

Department of Mathematics, University of Kelaniya, Kelaniya, Sri Lanka.