Own state

Eigenstate is a fundamental term in quantum physics . The eigenstate of a given physical quantity is a state of a physical system in which this quantity has a well-defined value. Only this value can be obtained as a measurement result if an error-free measurement of this variable is carried out on a system that is in its own state. It is also referred to as the eigenvalue with which the observed variable is present in the observed state, and the physical variable itself is referred to in this context as the observable . The eigenstate is often characterized by specifying the observable and its eigenvalue, e.g. B. by a quantum number , which is the sequential number of the eigenvalue in a listing of all possible eigenvalues of the observables.

The eigen-states of the Hamilton operator have a special meaning because they are the energy eigen-states or stationary states of the system described by this Hamilton operator. For example, a hydrogen atom is in its energetically lowest possible state when it is in its own state of energy with the (main) quantum number n = 1.

Overlay state

A system can (with a few exceptions) assume different eigenstates of the same observable. Then, according to the rules of quantum mechanics , all superposition states are available to the system in which different eigenstates are present at the same time, each with a certain probability amplitude . If only eigenstates with the same eigenvalue are superimposed, then the superposition state is also an eigenstate of the same observable with the same eigenvalue. The result of a measurement of these observables can therefore be clearly predicted. However, if eigenstates are superimposed to form different eigenvalues, each of these eigenvalues can appear as a result with a certain probability during a measurement .

In other words: unlike in classical physics , not all measurable quantities in quantum mechanics have a well-defined value in every state. Therefore, it is not always possible to predict the result of a corresponding (error-free) measurement with certainty. If, however, a measured variable has a well-determined value in a state, then the state is referred to as the eigenstate of this measured variable and its well-determined value as the respective eigenvalue. The measurement always gives the eigenvalue and leaves the system in the same eigenstate.

Measurement results of non-commutable observables

The observables, for which there are no common eigen-states, deserve special attention . If a measurement has been carried out for an observable, i.e. one of its eigenvalues has been obtained as a result, the system is then in the corresponding eigenstate for this eigenvalue. If this eigenstate of the first observable is not an eigenstate of the second observable, it is in any case a superimposed state of its eigenstates, namely with different eigenvalues. For a measurement of the second observable, the exact result cannot be predicted; it can be any of its eigenvalues that is represented in this superposition. In addition, if you just swapped the order of the measurements, the system would be in a different state afterwards. Such observables are not called interchangeable . A well-known example are the two observables for position and momentum of a particle .

Representation in mathematical formalism

In the mathematical formalism a state is represented by a vector in the Hilbert space , e.g. B. represents a wave function ; an eigenstate of an observable corresponding to one of the eigenvectors (or eigenfunctions ) of the observables. The observable is represented by a self-adjoint linear operator . When applied to the eigenstate, the result is the same eigenstate, multiplied by a scalar factor. This factor is the eigenvalue of the operator in question in this state.

The superposition of different states is represented by a linear combination of the relevant state vectors or wave functions, the coefficients of the individual components precisely indicating the probability amplitudes.

notation

If the operator has the eigenvalues , then the eigenvalue equation for the -th eigenstate is written as follows: ${\ displaystyle {\ hat {A}}}$ ${\ displaystyle a_ {1}, \, a_ {2}, \, a_ {3}, \, \ dots}$ ${\ displaystyle n}$ ${\ displaystyle | \ psi _ {n} \ rangle}$

{\ displaystyle {\ hat {A}} | \ psi _ {n} \ rangle = a_ {n} | \ psi _ {n} \ rangle}

Example: The solutions of the stationary Schrödinger equation

{\ displaystyle {\ hat {H}} | \ varphi \ rangle = E | \ varphi \ rangle}

are the eigenstates of the Hamilton operator , so that with the eigenvalues we have: ${\ displaystyle | \ varphi \ rangle = | \ psi _ {n} \ rangle}$ ${\ displaystyle {\ hat {H}}}$ ${\ displaystyle E_ {n}}$

{\ displaystyle {\ hat {H}} | \ psi _ {n} \ rangle = E_ {n} | \ psi _ {n} \ rangle}

meaning

If the examined system is in an eigenstate of the corresponding operator before a certain measurement , then the reliable result of this measurement is precisely the eigenvalue . However , if the system is in a state that is not its own state , the measurement result cannot be predicted with certainty. Each of the eigenvalues is then a possible measurement result, the probability for the result (if the states are normalized to 1) being given by (i.e. by the square of the magnitude of the component of the vector along ). The scalar product itself is also called the amplitude of the state in the state . ${\ displaystyle | \ psi _ {m} \ rangle}$ ${\ displaystyle {\ hat {A}}}$ ${\ displaystyle a_ {m}}$ ${\ displaystyle | \ varphi \ rangle}$ ${\ displaystyle {\ hat {A}}}$ ${\ displaystyle a_ {1}, \, a_ {2}, \, a_ {3}, \, \ dots}$ ${\ displaystyle a_ {n}}$ ${\ displaystyle \ vert \, \ langle \ psi _ {n} | \ varphi \ rangle \ vert ^ {2}}$ ${\ displaystyle | \ varphi \ rangle}$ ${\ displaystyle | \ psi _ {n} \ rangle}$ ${\ displaystyle \ langle \ psi _ {n} | \ varphi \ rangle}$ ${\ displaystyle | \ psi _ {n} \ rangle}$ ${\ displaystyle | \ varphi \ rangle}$

After a measurement, the system under investigation is then in that eigenstate of the operator in question whose eigenvalue agrees with the measurement result. This is known as state reduction. She represents u. a. make sure that repeating the measurement immediately shows the same result.

properties

The eigen-states of the same Hermitian operator , but with different eigen-values, are orthogonal : if , then .

{\ displaystyle a_ {n} \ neq a_ {m}}

{\ displaystyle \ langle \ psi _ {n} | \ psi _ {m} \ rangle = 0}

If a number of pairwise orthogonal eigenstates of the same Hermitian operator have the same eigenvalue, this is called -fold degenerate . Every linear combination of these eigenstates is then also an eigenstate with the same eigenvalue, altogether a -dimensional subspace of the entire state space. It is arbitrary which basis vectors one chooses in it. ${\ displaystyle g}$ ${\ displaystyle g}$ ${\ displaystyle g}$

${\ displaystyle g}$ indicates the statistical weight of the eigenvalue in quantum statistics . This is abbreviated, but imprecisely, often expressed in such a way that there are “exactly different states” for this measured value . This expression refers to the maximum number of linearly independent states among all eigenstates with the same eigenvalue, i.e. the dimension of the subspace. ${\ displaystyle g}$

In general, every (normalized) linear combination of state vectors is a possible state vector ( superposition principle ), often also called the superposition state. If eigenstates of a certain operator are superimposed, then the superimposing state is an eigenstate of the same operator if and only if the linear combination only contains eigenstates with the same eigenvalue.