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Mobile App Chapter 8: Problem 24
Show that the expectation value of an operator that can be written as the square of an hermitian operator is positive.
Short Answer
Expert verified
Because the expectation value of the square of an Hermitian operator is a sum of non-negative terms, it is necessarily non-negative.
Step by step solution
01
Definition of Expectation Value
Recall that the expectation value of an operator \( \hat{A} \) in quantum mechanics is given by \( \langle \hat{A} \rangle = \langle \psi | \hat{A} | \psi \rangle \), where \( | \psi \rangle \) is the state vector of the system.
02
Operator as Square of Hermitian Operator
Let us consider an Hermitian operator \( \hat{H} \). An operator that is the square of \( \hat{H} \) can be written as \( \hat{A} = \hat{H}^2 \).
03
Hermitian Property of Squared Operator
Since \( \hat{H} \) is Hermitian, it has real eigenvalues and its eigenstates form a complete orthonormal basis. This means that we can express \( | \psi \rangle \) as a linear combination of the eigenstates of \( \hat{H} \).
04
Expansion of \( | \psi \rangle \) in Eigenstates
Write \( | \psi \rangle \) as \( | \psi \rangle = \sum_n c_n | \phi_n \rangle \), where \( \{ | \phi_n \rangle \} \) are the eigenstates of \( \hat{H} \), with corresponding eigenvalues \( \{ h_n \} \), and \( c_n \) are the complex coefficients.
05
Expression for Expectation Value
Substitute the expansion of \( | \psi \rangle \) into the expectation value formula to get \( \langle \hat{A} \rangle = \langle \psi | \hat{H}^2 | \psi \rangle = \sum_{n,m} c_m^* c_n \langle \phi_m | \hat{H}^2 | \phi_n \rangle \).
06
Utilizing Eigenstate-Eigenvalue Equation
Since \( | \phi_n \rangle \) are eigenstates of \( \hat{H} \), we can use the eigenstate-eigenvalue equation \( \hat{H} | \phi_n \rangle = h_n | \phi_n \rangle \).
07
Simplification Using Orthonormality
Using the orthonormality of the eigenstates \( \{ | \phi_n \rangle \} \) (\( \langle \phi_m | \phi_n \rangle = \delta_{mn} \)), the expectation value simplifies to \( \langle \hat{A} \rangle = \sum_n |c_n|^2 h_n^2 \), since the cross terms where \( m \) does not equal to \( n \) vanish.
08
Concluding the Positivity of the Expectation Value
Since \( h_n^2 \) is always non-negative (as it is the square of a real number) and \( |c_n|^2 \) is non-negative (being the square of the modulus of a complex number), it follows that \( \langle \hat{A} \rangle \) is a sum of non-negative terms and thus must be non-negative itself.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Hermitian Operator
In the enigmatic realm of quantum mechanics, Hermitian operators are a cornerstone concept, underpinning much of the mathematical framework that describes physical systems. A Hermitian operator, often denoted by the symbol ‘hat’, such as \( \hat{H} \), possesses some fundamental properties that align beautifully with observable quantities in quantum mechanics.
First and foremost, Hermitian operators are linear, meaning their action on a sum of states is equivalent to the sum of their actions on each state individually. Additionally, these operators are self-adjoint, which implies that their eigenvalues — the values you ‘expect’ to measure for a certain property, like energy or position — are always real numbers. This aligns well with the concept that measurements in the physical world yield real values, not complex ones.
The repercussions of this are profound: a system described by a Hermitian operator has eigenstates that form a complete set, which allows any state of the system to be expressed as a combination of these eigenstates. This elegant feature allows physicists to describe the state of a system with a precise mathematical language and to predict the outcomes of measurements with certainty when the system is in one of these eigenstates.
First and foremost, Hermitian operators are linear, meaning their action on a sum of states is equivalent to the sum of their actions on each state individually. Additionally, these operators are self-adjoint, which implies that their eigenvalues — the values you ‘expect’ to measure for a certain property, like energy or position — are always real numbers. This aligns well with the concept that measurements in the physical world yield real values, not complex ones.
The repercussions of this are profound: a system described by a Hermitian operator has eigenstates that form a complete set, which allows any state of the system to be expressed as a combination of these eigenstates. This elegant feature allows physicists to describe the state of a system with a precise mathematical language and to predict the outcomes of measurements with certainty when the system is in one of these eigenstates.
Quantum Mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. At its heart, it challenges our classical intuitions, introducing concepts like quantization, probabilities, and entanglement. One of the quintessential tools in quantum mechanics is the use of linear operators to describe physical observables, such as position, momentum, and energy.
These operators act on wavefunctions, which themselves are mathematical constructs that encapsulate the probabilities of finding a particle in a particular state. When an operator acts on a wavefunction corresponding to its eigenstate, we obtain an eigenvalue, which corresponds to the measurable value of the physical property for the state.
The framework that quantum mechanics offers is non-deterministic in nature and instead relies heavily on statistical and probabilistic interpretations. This means that rather than giving a definite outcome, it provides the likelihood of observing possible outcomes. The cornerstone of this probabilistic nature is the eigenstate-eigenvalue equation, which connects the mathematical description of a system with the measurable outcomes of experiments.
These operators act on wavefunctions, which themselves are mathematical constructs that encapsulate the probabilities of finding a particle in a particular state. When an operator acts on a wavefunction corresponding to its eigenstate, we obtain an eigenvalue, which corresponds to the measurable value of the physical property for the state.
The framework that quantum mechanics offers is non-deterministic in nature and instead relies heavily on statistical and probabilistic interpretations. This means that rather than giving a definite outcome, it provides the likelihood of observing possible outcomes. The cornerstone of this probabilistic nature is the eigenstate-eigenvalue equation, which connects the mathematical description of a system with the measurable outcomes of experiments.
Eigenstate-Eigenvalue Equation
The eigenstate-eigenvalue equation is a vital piece in the puzzle that is quantum mechanics — it is the formula that connects theory with observation, allowing us to peek into the probable outcomes of measuring a system. In its essence, the equation tells us that when an operator corresponding to a measurable quantity acts on an eigenstate of that operator, the result is just the eigenstate scaled by a certain value, called the eigenvalue.
Mathematically, this is written as \( \hat{A} | \text{eigenstate} \rangle = \text{eigenvalue} | \text{eigenstate} \rangle \), where \( \hat{A} \) is an operator and the eigenstate and eigenvalue are denoted traditionally by \( | \text{eigenstate} \rangle \) and a corresponding scalar value. In practical terms, an eigenvalue represents the result you’d expect to measure - for instance, the energy of an atom, if the atom is in a state that is an eigenstate of the energy operator.
The beauty of the eigenstate-eigenvalue equation is its simplicity and power. Since any state in quantum mechanics can be expressed as a combination of these eigenstates, this equation becomes a fundamental tool that lets physicists understand the possible measurements that can arise from a quantum system, thus formulating a predictive model that can be tested with actual data from experiments.
Mathematically, this is written as \( \hat{A} | \text{eigenstate} \rangle = \text{eigenvalue} | \text{eigenstate} \rangle \), where \( \hat{A} \) is an operator and the eigenstate and eigenvalue are denoted traditionally by \( | \text{eigenstate} \rangle \) and a corresponding scalar value. In practical terms, an eigenvalue represents the result you’d expect to measure - for instance, the energy of an atom, if the atom is in a state that is an eigenstate of the energy operator.
The beauty of the eigenstate-eigenvalue equation is its simplicity and power. Since any state in quantum mechanics can be expressed as a combination of these eigenstates, this equation becomes a fundamental tool that lets physicists understand the possible measurements that can arise from a quantum system, thus formulating a predictive model that can be tested with actual data from experiments.
