Mathematics of Quantum Mechanics ii
Part 3: Quantum Mechanics
3.1 Mathematical Postulates of Quantum Mechanics
-
Quantum States: Quantum states are described by unit vectors in a complex Hilbert space. In the context of quantum mechanics, the state of a quantum system is represented as a vector in a complex - valued Hilbert space , where .
-
Quantum Measurement: The probability of measuring a system in a given state is given by (Born’s rule). After measurement, the wave - function collapses into the state . If we have a set of orthonormal basis states in the Hilbert space, and the system is in the state , the probability of obtaining the state upon measurement is calculated using this rule.
-
Quantum Operations: Quantum operations are represented by unitary operators on the Hilbert space, i.e., , where is the conjugate - transpose of and is the identity operator. Unitary operations preserve the norm of the quantum state, .
-
Multiple Quantum States: The Hilbert space of a composite system consisting of subsystems with Hilbert spaces and is given by the tensor product .
-
Observables: Physical observables are represented by the eigenvalues of a Hermitian operator on the Hilbert space. A Hermitian operator satisfies .、
- Common Conclusions:
- Quantum states are normalized, , to ensure that , where is the probability of measuring the system in the state as given by Born’s rule.
- Unitary operations satisfy , which means they preserve the inner product and the norm of quantum states.
3.2 New Notation: The Bra - Ket Notation
- Ket: represents a vector in the Hilbert space.
- Bra: represents the conjugate - transpose of . If , then , where is an orthonormal basis of the Hilbert space.
- Bra - Ket: , which is the inner product of the vectors and .
- Example: Given and , the inner product .
- Common Conclusions:
- The bra - ket notation simplifies the representation of quantum states and operations. For example, when calculating the probability of a measurement outcome using Born’s rule , the notation makes the operations more intuitive.
- The inner product is a scalar that represents the overlap between two quantum states. A larger magnitude of indicates a greater similarity between the two states.
3.3 Single Quantum State and the Qubit
- Qubit: The basic unit of quantum information, with two exclusive states and . These states form an orthonormal basis for the two - dimensional Hilbert space of a qubit, , , and .
- Quantum Superposition Principle: A quantum system can be in a superposition of and , represented as , where and .
- Example: A qubit in the state is in a superposition of and with equal coefficients. Here, .
- Common Conclusions:
- The coefficients and in the superposition state determine the probabilities of measuring the qubit in the or state. Specifically, and .
- The superposition principle is a fundamental feature of quantum mechanics, distinguishing it from classical physics. In classical physics, a system can be in only one state at a time, while in quantum mechanics, a qubit can be in a superposition of multiple states simultaneously.
3.4 Quantum Measurement
- Born’s Rule: The probability of measuring in the state is .
- Wave Function Collapse: After measurement, the state collapses to .
- Example: Given and , the probability of measuring in is .
- Common Conclusions:
- Quantum measurement is probabilistic, and the outcome is determined by the inner product of the measured state and the basis states. This is in contrast to classical measurements, which are usually deterministic.
- The wave - function collapse is a non - reversible process that occurs upon measurement. Once the measurement is made, the state of the system changes from the superposition state to one of the eigenstates of the measured observable.
3.5 Quantum Operations
- Quantum Operations: Represented by unitary matrices such that .
- Example: The Hadamard gate is a unitary matrix . Applying to : .
- Common Conclusions:
-
Unitary operations are essential for manipulating quantum states without altering their norm. They can be used to change the state of a qubit or a multi - qubit system in a way that preserves the quantum information.
-
Quantum operations can be combined by matrix multiplication. For example, if and are two unitary operations, then applying followed by is equivalent to applying the product . The order of operations matters because matrix multiplication is non - commutative in general.
-
3.6 Multiple Quantum States
- Tensor Product: The tensor product of two vectors and is . If and , then .
- Bell States: A set of orthonormal entangled states for two qubits.
- Example: The Bell state .
Entangled States and Separable States
Definitions:
-
Separable State: A state of a composite quantum system that can be written as the tensor - product of states of its individual subsystems. Mathematically, a two - qubit state is separable if it can be expressed as:
where and are states of the individual subsystems. -
Entangled State: A state of a composite quantum system that cannot be written as the tensor - product of states of its individual subsystems. Mathematically, a two - qubit state is entangled if it cannot be expressed as:
Examples:
-
Separable State Example: Consider the state:
where is the standard basis state and is a superposition state. This state is separable because it can be written as the tensor - product of and . -
Entangled State Example: Consider the Bell state:
This state is entangled because it cannot be written as the tensor - product of two single - qubit states.- To see this, assume for contradiction that is separable:
Expanding the right - hand side:
Comparing coefficients with , we get:
From and , either or , and either or . However, if or , then or cannot be . Similarly, if or , then or cannot be . Therefore, no such exist, and is entangled.
- To see this, assume for contradiction that is separable:
-
Common Conclusions:
- The tensor product allows the representation of composite quantum systems. For a two - qubit system, the Hilbert space is the tensor product of the two - dimensional Hilbert spaces of each qubit.
- Entangled states exhibit correlations that have no classical analog, making them a key resource in quantum information processing. For example, in quantum teleportation and quantum key distribution, entangled states play a crucial role.
3.7 Observables
- Observables: Represented by the eigenvalues of a Hermitian operator on the Hilbert space.
- Example: The Pauli - Z operator is a Hermitian matrix. The eigenvalues of are and , corresponding to the eigenstates and respectively. To find the eigenvalues, we solve the characteristic equation , where is the identity matrix.
Conclusions:
-
Hermitian operators have real eigenvalues, which correspond to the possible measurement outcomes. This is important because physical observables, such as energy or spin, are real - valued.
-
The expectation value of an observable in a given quantum state is , which represents the average result of many measurements of the observable on the state .
Common natation
- Ket: represents a vector .
- Bra: represents the conjugate - transpose of .
- Bra - Ket: represents the inner product of and .
Common Conclusion
-
Unitary Operator :
A unitary operator satisfies the condition:
where is the conjugate transpose of , and is the identity matrix. -
Hermitian Operator :
A Hermitian operator satisfies the condition:
This means that for any vectors and in the Hilbert space:
-
Identity Matrix :
The identity matrix is a square matrix with ones on the diagonal and zeros elsewhere:
-
Pauli Matrices , , :
The Pauli matrices are a set of three Hermitian and unitary matrices:
-
Hadamard Gate :
The Hadamard gate is represented by the matrix:
This operator satisfies . -
Bell States , , , :
The Bell states are four maximally entangled two - qubit states:
本文作者:归游
本文链接:https://www.cnblogs.com/guiyou/p/18705485
版权声明:本作品采用知识共享署名-非商业性使用-禁止演绎 2.5 中国大陆许可协议进行许可。
【推荐】国内首个AI IDE,深度理解中文开发场景,立即下载体验Trae
【推荐】编程新体验,更懂你的AI,立即体验豆包MarsCode编程助手
【推荐】抖音旗下AI助手豆包,你的智能百科全书,全免费不限次数
【推荐】轻量又高性能的 SSH 工具 IShell:AI 加持,快人一步